Prime spectra in non communicative algebra

Lecture Notes in
Mathematics
Edited by A. Dold and B. Eckmann

444

E van Oystaeyen

Prime Spectra in
Non-Commutative Algebra

Springer-Verlag
Berlin-Heidelberg • New York 19 75


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Prof. Freddy M. J. van Oystaeyen
Departement Wiskunde
Universiteit Antwerpen
Universiteitsplein 1
2610 Wilrijk/Belgium

Library of Congress Cataloging in Publication Data

Oystaeyen~ F
van, 1947Prime spectra in noneonmm~tative algebra.
(Lecture notes in mathematics ; 444)
Bibliography:
p.
Includes index.

1. Associative algebras. 2. Associative rings.
3o Modules (algebra) 4. Ideals (algebra) 5. Sheaves~
theor~j of. I. Title. If. Series: Lecture notes in
mathamatics (Berlin) ; 444.

QA3.L28

no. 444

[0J~251.5]

510'.8s [5~'.24]
75 -4877

AMS Subject Classifications (197'0): 14A20, 16-02, 16A08, 16A12,
16A16, 16A40, 16A46, 16A64,
16A66, 1 8 F 2 0
ISBN 3-540-07146-6 Springer-Verlag Berlin. Heidelberg. New York
ISBN 0-387-07146-6 Springer-Verlag New York • Heidelberg • Berlin
This work is subject to copyright. All rights are reserved, whether the whole
or part of the material is concerned, specifically those of translation,
reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks.
Under § 54 of the German Copyright Law where copies are made for other
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be determined by agreement with the publisher.
â by Springer-Verlag Berlin ã Heidelberg 1975. Printed in Germany.
Offsetdruck: Julius Beltz, Hemsbach/Bergstr.


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CONTENTS

Introduction

1

1.

Generalities on Localization

4

1.1.

Kernel Functors

4

1.2.

L o c a l i z a t i o n at a Prime Ideal

8

II.

Symmetric Kernel Functors

16

11.1.

Localization at Symmetric Kernel Functors

16

11.2.

Q u a s i - p r i m e Kernel Functors

24

11,3.

Reductions

31

III.

Sheaves

42

111.1. Spec and the Zariski Topology

42

111.2. Affine

48

Schemes

IV.

Primes in Algebras

IV,1.

Pseudo-plaees

IV°2.

S p e c i a l i z a t i o n of Pseudo-places

62

IV.3,

P s e u d o - p l a c e s of Simple Algebras

67

IV.4.

Primes in A l g e b r a s over Fields

71

IV.5.

Localization at Primes,

81

V_~.

Application

V.1.

Generic Central Simple Algebras

86

V.2.

Two Theorems on Generic

9O

V.3.

The Modular Case

VI.

over Fields

of Algebras

over Fields

and Sheaves

: The Symmetric Part of the Brauer Group

Crossed Products

58
58

86

96

__Appendix : L o c a l i z a t i o n of A z u m a y a Algebras

101

VI.1.

The Center of Qo(R)

101

VI.2.

L o c a l i z a t i o n of A z u m a y a Algebras

109

VI.3. A z u m a y a Algebras
VI.4.

over Valuation Rings

Modules over A z u m a y a Algebras

References

for the A p p e n d i x

References
Subject Index

115
119
122
123


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LEITFADE,N

IVy- ................

II

/

"-- I I I

V

VI


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ACKNOWLEDGEMENT

The author is indebted to Professor Dave Murdoch
at the U n i v e r s i t y of British C o l u m b i a for reading
the m a n u s c r i p t and for many helpful suggestions.
Part of this r e s e a r c h was done at Cambridge University and I am thankful to the people of the mathematical department for the h o s p i t a l i t y enjoyed there.
I was able to continue with this work while serving
in the Belgian army and I thank my superiors

for

not trying too hard to make a soldier out of a mathematician.
I am obliged to the U n i v e r s i t y of Antwerp for facilities and the necessary financial support.
I thank Melanie for reading the illegible and for
careful typing.

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SYMMETRIC L O C A L I Z A T I O N AND SHEAVES

Introduction

Present notes are mainly concerned with two topics
zation and p s e u d o - p l a c e s of algebras over fields.

: symmetric locali-

The first three sec-

tions deal with localization theory while in the remaing sections the
accent is on pseudo-places.

This split up is r e f l e c t e d

in a shift of in-

terest from prime ideals to completely prime ideals.

A f t e r a brief summary of the basic facts about kernel functors,

loca-

lization techniques e x p o u n d e d by P. Gabriel and 0. Goldman are adapted,
in Section i!, so as to yield a s a t i s f a c t o r y ideal theory.

In Section

III we construct a presheaf of n o n c o m m u t a t i v e rings on the prime spectrum,
Spec R, of a left N o e t h e r i a n ring.

If

R

is a prime ring then this pre-

sheaf is a sheaf and Spec R is then said to be an affine scheme.
though Spec is not n e c e s s a r i l y functorial,
cause of the many local properties
is an ideal of

R

studying it is worthwile be-

that still hold.

For example,

where QA(R)

if A

such that the Zariski open subset X A of Spec R is such

that the a s s o c i a t e d localization functor QA has property

by 0. Goldman,

Al-

(T) discussed

then X A is an affine scheme too, in fact X A ~ Spec QA(R),

is the ring of quotients with respect to QA"

Local proper-

ties like this are related to the question whether the extension of an
ideal

A

of

R

to a left ideal Qo(A) of Qo(R), for some symmetric kernel

functor o, is also an ideal of Qo(R).

Therefore the local properties

of the sheaf Spec R lean heavily on the ideal theory e x p o u n d e d in II. 1.

The r e l a t i o n between symmetric localization at a prime ideal of a

left N o e t h e r i a n ring and the J. Lambek - G. Michler torsion theory,
[21], is explained in II. 2.
functor o R _ p at a prime ideal

cf.

It turns out that the symmetric kernel
P

of

R

is the biggest symmetric kernel


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functor
spired

smaller

than the J. Lambek

by this we define

be the biggest

Goldman's

supporting

quasi-prime
functor
prime

left

prime

ideals

prime

ideal

some

special

Using

functor

module.

The

modules.

sense

ideal.

is strongly

to be natural

should

functor

ideal while

cases,

indeed

correspond

the c o r r e s p o n d e n c e

prime

ideals

A,

subrings

of the k - a l g e b r a

of a place

of a field.

specialization

applies

fields,

the c o m m u t a t i v e

over

The general

fields,

which

thus

theory

us to construct

in the n o n c o m m u t a t i v e

stalks

of Primk(A)

case.

there

is added

are several

symmetric

about

Interrelations

between

the

a sheaf Primk(A)

functors
these

sheaf
we con-

concept

and their

primes

in alge-

on

A.

but this

functors
associated

to l o c a l i z a t i o n

kernel

in

are ring h o m o m o r p h i s m s

The kernel

interest

to every

Therefore

of p s e u d o - p l a c e s

turn out to be the q u a s i - p r i m e s

between

of the

generalizing

information

kernel

completely.

case Prim k is a functor Al~k ~ Sheaves,

true

ted to a prime.

Connell.

so as to yield

allowing

always

However

by I.G.

a

and moreover,

may be d e s c r i b e d

to

of a

one because

a quasi-prime

A,

related

then be the analogue

we get a g e n e r a l i z a t i o n

introduced

induced

a prime

is the better

does

In-

to c o n s i d e r

functors

of certain

IV.

2., to

and q u a s i - p r i m e
there

there

in II.

that the c o r r e s p o n d e n c e

of algebras

cause

functor,

be noted

s~der p s e u d o - p l a c e s

ideals.

~p.

It should

completely

over

theory

concept

of a prime
[12]

torsion

than the kernel

latter

It seems

of Goldman

GamkA on a k - a l g e b r a

bras

kernel

smaller

as a " g e n e r a l i z a t i o n "

in the

Michler

a quasi-prime

symmetric

by a q u a s i - s u p p o r t i n g

-G.

In

is not

at the
to prime

at primes

be-

which may be associa-

are the main

subject

of

5..

A further
Section

V.

generic

of the theory

To every finite

from fields
galoisian

application

and surjective

pseudo-places.

property

:

abelian
places

The

of p s e u d o - p l a c e s

group

G

to crossed

skew field

is given

we construct
product

Dk(G)

(Dk(G)

in

a functor

skew fields

and

= ~(G)(k))

has

~(G)

a


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every crossed product

(G,I/k,{Ca,T}) , where

i/k is an abelian ex-

tension with Gal(i/k)

m G, defined by a symmetric

{C ,TIo,T e G}, is residue algebra of ~k(G)
do-place.

In this way a p a r a m e t r i z a t i o n

factor set

under a galoisian

of certain

subgroups

pseuof

the Brauer Group is obtained.

In the Appendix,

Section VI, the theory of symmetric

tion is applied to an Azumaya algebra

be proved there that symmetric
cended"

to kernel functors

Moreover,
properties

localization

R with center

kernel functors

localiza-

C.

It will

on M(R) may be "des-

on M(C) when they have property

at a prime ideal

P

of

one could hope for and therefore

R has all the good

Spec R is as close as

one can get to an affine sheaf in the commutative

case.

The J. Lambek,

coincides

the symmetric
bra,

G. Michler torsion theory Op at
°R-P in case

and Op has property

symmetric

kernel functor

tral extension

of

R,

R

(T).

The ring of quotients

of Azumaya algebras

ple algebras.

algeat a

then every ideal

is an ideal of Qo(R).

over valuation rings

related to the theory of unramified

Qa(R)

with

algebra and it is a cen-

so if o is a T-functor

A ~ T(o) has the property that Qo(A)
Localization

P

is a left Noetherian Azumaya

o is an Azumaya

(T).

pseudo-places

is closely

of central

sim-


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I. G E N E R A L I T I E S

I.

1. K e r n e l

Functors.

All rings

considered

mean

left m o d u l e

sided

ideal.

les.

A functor

following

Let

R

c(M)

= M,

defined
free

Denote

submodule

g(M)

N

is a s u b m o d u l e

f(~(M))

of

M

c ~(N)

we h a v e

it is G - t o r s i o n

free

if ~(M)

the

class

A kernel

of t o r s i o n

functor

a filter

is a G - t o r s i o n

has

the

following

If A e T(~)

T(~)

module.
properties

and

Ideal

the

Module

stands

category

functor

will

for t w o -

of R - m o d u -

if it h a s

the

if

B

M.

.
= N n a(M).

is said to be g - t o r s i o n
= 0.

A torsion

objects

~ on M(R)

= 0 for all M • M(R).

of

a(N)

M • M(R)

R/A

by M(R)

is a k e r n e l

then

is a s s o c i a t e d

element.

be u n i t a r y .

functor

objects.

B e

to M(R)

a unit

:

M • M(R),

by g i v i n g

~(M/~(M))

2.

be a ring.

a f r o m M(R)

Let a be a k e r n e l

1.

will

If f • H o m R ( M , N ) , t h e n

3. For a n y

to h a v e

all m o d u l e s

properties

1. For e v e r y
2.

and

are a s s u m e d

ON L O C A L I Z A T I O N

and the

is c a l l e d

To an a r b i t r a r y

consisting

of left

The

T(~),

filter

theory
class

is in fact
of t o r s i o n

idempotent

kernel

ideals

sometimes

if

if

functor

A

of

R

called

~ there

such that

a topology,

:

is a l e f t

ideal

of

R

such that A c B then

T(a).

If A , B • T(~)

t h e n A n B • T(~).

3. For e v e r y A • T(a)

and a n y x • R t h e r e

exists

a B • T(a)

such

that

B x c A.

4.

Let M • M ( R ) ,

then

x • g(M)

if and o n l y

if t h e r e

is an A ~ T(a)

such

t h a t A x = 0.

Any

filter

ve,

defines

Conversely,

T

of l e f t

a topology
to s u c h

ideals
in

R

a filter

of

R,

such
T

having

that

there

R

properties
becomes

corresponds

1,2,3,

listed

a topological
a functor

abo-

ring.

~ on M(R),


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defined

by o(M)

= {x • M, A x

funetor with T(o)
functors
talk

the

a topology

set F(R)

of kernel

N

of

This t o p o l o g y

M

for which

is called

F(R) may be p a r t i a l l y
for all M • M(R).

the quotient

induced

ordered

Gathering

PROPOSITION

1.

Equivalently

1. o • F(R)

is idempotent.

then

M

N

by the o - t o p o l o g y
4. If B c A are left
torsion,

Remark.

then

being

in

M

R

A o • F(R)

M/N

to

induces

of 0 in

M

the

is o-torsion.

is exactly

T(o).

o < T if and only

from

and

[12] we obtain

The

set

if o(M) c T(M)

the following

if both

in

coincides
of

K

and

I

are o - t o r s i o n

mo-

R

M,
with

then

the topology

the o - t o p o l o g y

such that A • T(o)

in

induced

in N

N.

and if A/B

is o-

B • T(o).

If o • F(R)

is idempotent,

then T(o)

is a m u l t i p l i c a t i v e l y

clo-

sed set.

An I e M(R)

is said to be o - i n j e c t i v e

0 ~ K ~ M ~ M/K ~ 0 with M/K being
there

is an f • HomR~M,I)

A o-injective
f to

M

module

verified

to be f a i t h f u l l y
If

I

I

if, for every

o-torsion

extending

exact

sequence

and any f e HomR(K,I) ,

f to M.

is f a i t h f u l l y

o-injective

if the e x t e n s i o n

T of

is unique.

It is easily

:

module.

o-open

ideals

it p o s s i b l e

:

is a o - t o r s i o n

3. Let M,N • M(R),

in

results

module

kernel

in M.

by putting

2. If 0 ~ K ~ M ~ I ~ 0 is exact
dules,

between

makes

on M(R).

o is a kernel

for the n e i g h b o r h o o d s

the o - t o p o l o g y

M = R, the o - t o p o l o g y

type,

functors

taking

This

correspondenee

of the p r e s c r i b e d

in every M • M(R)

submodules

for some A • T}.

The o n e - t o - o n e

and topologies

about

Taking

= T.

= 0

that a n e c e s s a r y

o-injective

is o - i n j e c t i v e

is that

then every

and sufficient
I

is o-torsion

f • HomR(A,I)

condition

for

I

free.

with A • T(o),

extends

to


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an ~ • HomR(R,I) ; this condition is clearly also a sufficient one.

Unless otherwise

specified, o will always be an idempotent kernel

functor from now on.
If M • M(R)

is o-torsion free then there exists a faithfully o-injec-

tive I • M(R), containing

M,

then unique up to isomorphism,
of

M,

X
I

of

E

E

with the property o(X/M)

containing

M

This

I

is

it can be c o n s t r u c t e d as the extension

in some absolute injective hull

dules
tive

such that I/M is G-torsion.

of

M,

= X/M.

maximal among submoThe f a i t h f u l l y o~injec -

will be denoted by Qo(M).

The d e f i n i t i o n of Qo(M)

may be considered as the direct limit of the system

{H°mR(A'M)'
where TAB(f)
If

M

~A,B

: H°mR(A'M) ~ H°mR(B'M)' A n B • T(o)},

=f]B.

is not o-torsion free then we put Qo(M)

= Qo(M/o(M))

and the di-

rect limit i n t e r p r e t a t i o n yields at once that Qo is a covariant and left
exact functor on M(R),

(if the R-module

defined in the usual way, cf.
ning R/o(R)

as a subring.

[12]).

structure on the direct

Moreover,

limit is

Qo(R) is a ring contai-

The ring structure of Qo(R)

dule structure of Qo(R) and it is unique as such.

induces the R-mo-

The ring Qo(R) toge-

ther with the canonical ring h o m o m o r p h i s m j : R ~ Qo(R), provides us
with a satisfying localization technique.
of Qo is not garanteed.

P R O P O S I T I O N 2.

Recall from

[12]

In general, right exactness
:

The following statements are equivalent

1. Every M • M(Qo(R))
2. For all A • T(o),
3. Every M • M(Qo(R))
4. For all M • M(R),

:

is o-torsion free.
Qo(R)j(A)

= Qo(R).

is faithfully o-injective.
Qo(R) ® M m

Qo(M).

R
5. The functor Qo is right exact and commutes with direct sums.


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Let

a •

R-module
dules
map

F(R)

P

M,M'

sion

rows
P

sequence

cf.

,p

M'

~M

projective
[12],

that

in P r o p o s i t i o n

Note

contains
h

ideal

for

An

of Qa(R)

Noetherian

that,

then

a B • T(a)

0

The

second

A • T(a),

then

which
the

L.

Silver

one

property
becomes

of the

is g e n e r a t e d
Q~(R)

with

that

that

free

R-mo-

an R - l i n e a r

P/P'

the

An

is ~ - t o r -

diagram

:

if and o n l y

is left

F(R)

then

properties

having

one

a T-functor.

by a left

It is

if e v e r y

Noetherian

equivalent

a •

Qa is

listed

of the

In this

ideal

it is s u f f i c i e n t

is a - p r o j e c t i v e

~M

since

is a - p r o j e c t i v e .

proper-

case

of R / a ( R ) ,

so if

is too.

commutative

because,

diagram

that
from

every

A • T(a)

a morphism

:

~A

• 0

B' • T(a).

in P r o p o s i t i o n

trivial

R

2 is c a l l e d

~ B' --.---~ B c

is a - t o r s i o n

for a T - f u n c t o r

such

exact

if

idempotent

M'

A/B'

a-torsion

such

it c e r t a i n l y

a to be a T - f u n c t o r

: A ~ M we d e d u c e

where

P

functor.

~0

if any

ties

R is left

P' ~ M'

However,

2 holds.

left

: given

P' of

Qa is r i g h t

in P r o p o s i t i o n

every

if

kernel

is c o m m u t a t i v e .

if and o n l y

mentioned

map

,p'

is a - p r o j e c t i v e .

exact

idempotent,

M' ~ M ~ 0 t o g e t h e r

is a s u b m o d u l e

is an R - l i n e a r

is a b s o l u t e

A • T(a)
right

there

exact,

well-known,

necessarily

to be a - p r o j e c t i v e

an e x a c t

then

and t h e r e

with
If

is said
and

P ~ M,

be a, not

under

2 states

extension

that

every

to the r i n g

left

ideal

of q u o t i e n t s

Qa(R)

a.

started

investigating

the

correspondence

between

prime

ideals


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of

R

not

in T(o)

in g e n e r a l ,

one

these

sets

from

example

of rings

this

tion

8 of

where

2.

in s e c t i o n

a cross-cut
A.G.

i.e.,

In the

The

R

i.e.

functors

may

of t o r s i o n

D.C.

at a Prime

an ideal

sequel,

does

hold;

R - P.

linked

In the

be a s s o c i a t e d

Murdoch

used

may

be d e d u o e d
a class

we r e t u r n

ring,

to the

to

locali-

localiza-

noneommutative
to a p r i m e

by A.

and the

that,

between

is to d e t e r m i n e

way

theories

shown

this

is a c o m m u t a t i v e

Goldie,

case

ideal.
J.

Lambek,

author.

Ideal.

P

of

R

is p r i m e
there

If R - P is a m u l t i p l i c a t i v e
ideal

of fact

correspondence
In case

been

correspondence

question

is in a n a t u r a l

Heinicke,

prime

if and

only

if R - P is an

is an x • R such
set t h e n

P

that

is said

to be a

.

R will

always

be a left

Noetherian

ring,

unless

other-

specified.

To a p r i m e

ideal P of

R

the m u l t i p l i c a t i v e

: {g e R, r ~ P i m p l i e s

important
Define

part

Op e

that

Michler
the

47).

if Sl,S 2 • R - P t h e n

s l x s 2 e R - P.
completely

a one-to-one

system,

kernel

Localization

m-system~

such

P

It has

as a m a t t e r

II.

ideal

By d e f i n i t i o n ,

G(P)

p.

of Qo(R).

[31],

the o n e - t o - o n e

several

G. M i c h l e r ,

wise

(cf.

at a m u l t i p l i c a t i v e

We p r e s e n t

ideals

establish

[12],

at a p r i m e

however,

I.

cannot

of ideals,

problem

zation

and p r i m e

in G o l d i e ' s

F(R)

by its

injective

that
hull

HomR(M,

E(R/P))

jective

module

: 0).
E

T(Op)

# ~ for

Op c o i n c i d e s
E(R/P)

as

is a s s o c i a t e d .

localization

filter

[A : r] n G(P)

proved

rg ~ P},

theory,

r • R.

O. G o l d m a n
follows.

defines

In

[21],
functor

(M • M(R)
a kernel

For any M e M(R),

T E ( M ) = r~ {Kernels

of R - l i n e a r

[10],

of left

the k e r n e l

of R/P • M(R),

This

of.

consisting

every
with

set
an

[11].

ideals
J.

A

of

Lambek,

determined

is t o r s i o n
functor

put

maps

set p l a y s

M ~ E}.

R

G.
by

if

T E by an

in-


www.pdfgrip.com

Hence,
for

TE-torsion

left N o e t h e r i a n

TR/P

are

rings

tative

way
ring

F(R)

contains
is said

is the

following

= 0 and this

Op c o i n c i d e s

is c a l l e d

an ideal

R

to be s y m m e t r i c

3.

which

shows

that

with

closed.

PROOF.

implies

sely,

if A • T(o)

cient

to p r o v e

with

and

that

is p o s s i b l e
Since

because

C,A • T(o)

Transfer

of the

for k e r n e l

for all ~ • V and

{or,

and

o(A/B)

of e l e m e n t s

= ~ o

the

ideals
= A/B.
that

(M),

that

• V, t h e n

v

o(A/B)

R

is left

C.l • T(o)

if e v e r y

A • T(o)

functor

This

= A/B

o

then

that

it is suffiA = Ra I + . . . + R a n

C = n C. we o b t a i n
i
l
• T(o) h e n c e B • T(o).

CA

for f i l t e r s y i e l d s
For a set

{Or,

kernel

filter

a partial

Obviously

~ ~ o.

filter

functors

is a s s o c i a t e d

if p • F(R)

{or,

we d e f i n e

o ~ ov

kernel

functors

is s y m m e t r i c .
and w r i t e

by all

finite

~ • V}.

and

Now

T(~)

for

products

to a s y m m e t r i c

is i d e m p o t e n t

sup of

ordering

It is i m m e d i a t e

functors

generated

C A c B.

v • V} c F(R)

M • M(R).

kernel

Conver-

C.l a.l C B, w h i c h

inf of a set of i d e m p o t e n t

o is the

if T(o)

set.

Noetherian,

such

for all v • V t h e n

the

if and only

Putting

for e v e r y

be the

that,

p ~ o; this

only

ker-

of ideals.

A bilateral

that

inf of s y m m e t r i c

Let T(o)

property

an i d e m p o t e n t

is a m u l t i p l i c a t i v e

Since

ordering

if T ~ o

in u T(~).

if and

in a n o n c o m m u -

a basis

is s y m m e t r i c

T(o)

(see before).

the

has

in T(o).

v • V} be a set of s y m m e t r i c

the filter T(Ov).

o with

that

Choose

definition

is i d e m p o t e n t

F(R)

B c A is such

inclusion

i~f o v by o(M)

this

o •

it f o l l o w s

functors

R - P,

filter

functor

is also

B • T(o).

a 1 , . . . , a n • A.

its

ideal

if it is i d e m p o t e n t .

A bilateral

Idempotency

that

a bilateral

of

at a prime

: to the m - s y s t e m

such

is m u l t i p l i c a t i v e l y

let

functor

localization

is a s s o c i a t e d

PROPOSITION

from

by H o m R ( M , E )

the k e r n e l

of i n t r o d u c i n g

functor

A o •

o+=

given

= TE(R/p).

Another

nel

modules

functor

p ~ o v for

all


www.pdfgrip.com
10

Now,
ted.

to an m - s y s t e m

Let M • M(R)

The

topology

of

R

T(R-

containing

in the

absence

bilateral
closure

but

JR-

R- P a symmetric

and

define

P) c o r r e s p o n d i n g
an

ideal

of the

P'

sense

p(R)

ponent
ideal

and

o R _ p(R)

of t h e
A

of

zero

R

of G o l d m a n

but

we m a y

as d e f i n e d

associate

in

for

In t h a t

[12], m a y

s R m c N,

= 0 for
of t h e

R,

some
left

the

symmetric.

by

kernel

Note

lower

Finally,

that,

idempotent

s • R- P yields

[24].

ideals

o R _ p is s t i l l

case

and

s • R-P}.

Observe

be d e f i n e d

the u p p e r

symmetric

o R _ p is a s s o c i a -

s • R - P.

condition

not n e c e s s a r i l y

are r e s p e c t i v e l y

ideal

by some

idempotent.

p(M) = n {N, N c M,

T h e n J R - P is i d e m p o t e n t

sRm

to o R _ p c o n s i s t s

left N o e t h e r i a n

in the

functor

= {m • M,

(s) g e n e r a t e d

not n e c e s s a r i l y

~R-

~R-

o R _ p(M)

kernel

:

m • N}.

also

that

(R- P)-com-

to an a r b i t r a r y

functors

o A a n d A o,

i.e.,

oA

and both
sible

depend

d i n g to t h e
metric
and

since

o 0 is
Op a n d

T(o)

the

r • R,

only
F(R)

The

: r] n G(P)

is e q u i v a l e n t
entailing

define

A o = sup{o R_P,

of

will

A

in

be u s e d ,

o 0 as b e i n g

upon

the

smaller

ideals

than

relation

o.

the

Though

Indeed,

kernel

o R _ p c a n be e x p r e s s e d

are p l a u -

III).

functor

For

correspon-

o D is the b i g g e s t

o 0 is c l e a r l y

it f o l l o w s

the

both

(in s e c t i o n

in T ( o ) ;

closed

between

R.

P n A}

bilateral

immediately

Lambek-Michler
as f o l l o w s

sym-

that

torsion

theory

:

Op0 = OR _ p.

If A • T ( o ~ )
[B

first

is m u l t i p l i c a t i v e l y

symmetric

4.

the

based

functor

symmetric.

PROPOSITION

PROOF.

o •

filter

kernel

p n A],

o n l y o n the r a d i c a l

definitions,

an i d e m p o t e n t

= inf{o R_P,

with

then
# ¢.

B n G(P)

0
Op ~ o R _ p.

A

contains
Since

B c

# ¢, h e n c e

Moreover,

an i d e a l
[B

: r]

B • T(Op),
it f o l l o w s

B • T ( R - P) a n d

if x • o R _ p ( R / P )

then

i.e.
that

for e v e r y
B • T(Op)

a l s o A • T ( R - P),
C x

c p for

some


www.pdfgrip.com

11

ideal
this

C e T(Ryields

P) a n d

some

x 6 p and ~

o R _ p ~ TR/P.

Now,

x • ~.

Thus

= O, i m p l y i n g

because

TR/P

sRx c p for

that

= Up,

some

o R _ p(R/P)

and

s • R- P and

= 0 which

o R _ p is s y m m e t r i c

yields

we g e t

0

o R _ p < Op.

Symmetric
play

kernel

the m a i n

me-spectrum
paid

left

ideals

noted

left

left

ideals

5.

PROOF.

"if" p a r t

(section

symmetric
(T),

due

of

for

R

for a f i x e d

sheaf

to t h e

The

that

idempotent

is the

principal

in T ( R - P).

closed,

if it is

if it is m a x i m a l

idempotent

o •

kernel

pri-

price

theories

fact

or s i m p l y

funtors

on t h e

III).

torsion

is c r i t i c a l

some

these

not n e c e s s a r i l y

is o - c l o s e d ,

The

A is a c r i t i c a l

the R - m o r p h i s m
o(R/A)

hence
o.

A

ideals

ring

s 6 R - P are
R

f.i.

F(R).

The

funetor

among

set of

o will

be de-

by C'(o).

PROPOSITION

onto

of

ideal

prime

property

advantages,

of a s t r u c t u r e

of u s i n g

by some

ideal

A left

o-closed

critical

advantages

a left

several

construction

Noetherian

generated

in T(o).

proper

have

of i n v e s t i g a t i n g

say t h a t

not

in the

of a left

f o r the m a n y

difficulty

We

role

funetors

B

is t h e n

it is a l s o

~.

R/B

is u - t o r s i o n ,

A • C'(o),

TR/A(R/A)

B contains

A

properly

while

A

B

since

that

R-submodule
contains

for

some

A

then

Consider
R

mapped

properly

idempotent

= R/A.

yields

and

functor

: (R/A)/o(R/A)
The

that

is T R / A - C l o s e d .

B ~ T(TR/A)

of

R/B ~ (R/A)/(B/A)

: 0 and t h i s
A

if A e C ' ( ~ R / A ) .

is G - c r i t i c a l .

R = B and o(R/A)

o(R/A)

= 0, we h a v e

if and o n l y

be the

A e C'(o)
but

hence

thus

B

# 0 then

because

free,

ideal

Suppose

let

If o ( R / A )

in T(o)

u-torsion

contradicts

is t r i v i a l .

: R ~ R / A and

under

Therefore

Because

~

left

latter

o < TR/A.

Moreover,

B 6 T(o)

if

c T ( T R / A ) , con-

tradiction.

Let
R-linear

A

be o - c r i t i c a l
map

~s

: R/[A

and

let

s

: s] ~ R/A,

be an e l e m e n t
defined

not

by x m o d [ A

in

A,

then

: s] ~ x s

the
mod A,


www.pdfgrip.com
12

is a monomorphism.

It follows

the p r o p e r t y

= R/B and hence

in T(o),
cible
sI

left

~ A,

lated

or

o(R/B)

[A : s] • C'(o).
ideals

A

B

s 2 ~ B such that

if and only

Critic a l

and

left

if the

left

class

ideal

of

A

ideals

PROPOSITION
are

of related

6.

be a prime

P

among

A n G(P)

[8], that

I(R/B)

irredu-

if there
A

and

are

B

are re-

isomorphic.

as being the m a x i m a l

prime

left

ideals.

then A •

elements

This

in its i d e a l i z e r

if x • R A - A

exist

implies~

A R in R,

[A : x],

and by

class we get A : [A : x] and the
set follows

ideal

P

of

R

ideal

of

R.

easily.

Critical

are of p a r t i c u l a r

The

following

prime

interest.

statements

left

ideals

prime

not

left

intersecting

ideal

of

R

G(P).

containing

P

and

prime

left

ideal of

R

containing

P

and

= ~.

and r e l a t e d

properties

may be c o n n e c t e d

to this.

the prime

ideal

P

ning

Proposition

blem because

of

R;

may be found

in [21].

Let o = o R _ p be the
characterize

8 fails

between

The f o l l o w i n g

symmetric

the o - c r i t i c a l

to give a s a t i s f y i n g

the c o r r e s p o n d e n c e

problem

localization

left

solution

ideals

for this

C'(o R _ p) and C'(TR/P)

prime

of a "critical
kernel

left

funetors,

ideal"

cf.

[12].

is strongly

connected

at

contaipro-

is not

enough.

The concept
Goldman's

and

ideal not

= ~.

3. A is a critical

known well

V. Dlab

left

: s] has

:

2. A is an i r r e d u c i b l e

P.

Indeed

a prime

1. A is m a x i m a l

This

cf.

irreducible

is c o m p l e t e l y

containing

equivalent

A n G(P)

also,

I(R/A)

is a m u l t i p l i c a t i v e

Let

of R/[A

[A : s 1] = [B : s 2] and that
injectives

A

B

are said to be r e l a t e d

in the e q u i v a l e n c e

fact that A R - A
left

R

submodule

[A : s] is a m a x i m a l

Recall
of

with A R = {x • R, Ax C A}.
maximality

every

ideals may also be c o n s i d e r e d

in an e q u i v a l e n c e
a critical

that

with

Let o • F(R) be idempotent.


www.pdfgrip.com
13

A support
nonzero

for

o is a o - t o r s i o n

submodule

is c a l l e d

a prime

such that

T S = o.

Clearly,

if

S

S is a n y
(up to

S

kernel

functor

free

support

ideal

we h a v e

7.

for o t h e n

in R, t h e n

that

has

to be

be

such

a support

nonzero

injective.

Moreover,

A ~ e
S

exists
time

let

A

F(R)

for o

If o is p r i m e

same

and

for every

homomorphism

there

at the

idempotent

that,

is o - t o r s i o n .

exists

any

for o which

F(R)

S

S/S'

if t h e r e

T S = o.

support

Let o e

R-module

f o r o, t h e n

R-module

isomorphism)

PROPOSITION
left

S' of

is a s u p p o r t

to a o - t o r s i o n

free

from
and

S
if

a unique

is o - i n j e c t i v e .

be a o - c r i t i c a l

:

1. A is T R / A - c r i t i c a l .
2. T h e

quotient

3. T h e

induced

module
kernel

PROOF.

The

nonzero

submodule

A properly

The

last

first

and

o = inf{TR/A,

PROOF.

The

of R/A

A ~ e

is prime.

follows

from proposition

of some

it is a - o p e n

F(R)

and

immediately

is

f o r o.

left
thus

from

idempotent

ideal
R/A

1 and

5.
of

Secondly,
R

which

is a s u p p o r t

every

contains

f o r o.

2.

if and o n l y

if

A • C'(o)}.

fact

Then

TR/A

is i m a g e

follows

that

for e v e r y A • C'(o)
T ~ o.

functor

as such,

8.

is a s u p p o r t

statement

statement

PROPOSITION

R/A

there

o(R/A)

a n d thus

= 0 f o r a n y A e C'(o)
o < inf{TR/A,

is a C • T(T)

A 1 • C'(o)

such

that

C c A 1.

T ( TR/A1

cannot

hold,

hence

converse,

define

C'(O)

to be the

R maximal

in an e q u i v a l e n c e

T(o).

implies

A e C'(o)}.
Since

C ~ T(o)

that

Let T ~ a w i t h
we m a y

For t h i s p a r t i c u l a r A 1 it f o l l o w s
o = inf{TR/A,

class

A • C'(o)}.

set of m a x i m a l
of r e l a t e d

o ~ TR/A

find

an

that

To p r o v e

the

o-closed

left

ideals

irreducible

left

ideals.

of


www.pdfgrip.com
14

Since ~R/A is idempotent for any A e C'(o), o is idempotent too.

COROLLARIES.

If o • F(R) is idempotent then o = T M where

M

is the di-

rect sum of the n o n - i s o m o r p h i c quotient modules R/A for all A • C'(o).
Furthermore,

A • C'(a).

o = T N where N = ~ Qo(R/A), the direct sum ranging over all
It is clear that M (or N) cannot be a support for o if there

exist at least two factors in the sum, whence the following results.

An

idempotent o • F(R) is a prime kernel functor if and only if Q (R/A) ~ E
for all A • C'(o).
A l t e r n a t i v e ways of looking at critical left ideals are e n c o u n t e r e d in
[19],

[32]; they may be described as left annihilators of the elements

of i n d e e o m p o s a b l e

injective modules,

so they are related to what

is cal-

led an atom in [32].
For completeness sake,
for

R

let us recall that the left A r t i n i a n c o n d i t i o n

is equivalent to every critical prime left ideal being a maximal

left ideal of

R.

A r t i n i a n conditions will be avoided in the present

context.

The c o r r e s p o n d e n c e between prime ideals of

R

Qo(R) has been studied in case o : ap in [21],
get useful
Let

P

Then

pect to G(P)

R

In order to

R,

Then G(~)

The image of

P

set G(P)

under R ~ R/Op(R) will be denoted

= (G(P) + Op(R))/Op(R),
R

From

and by s t r a i g h t f o r w a r d argumen-

satisfies the left Ore condition with respect

if and only if R/Op(R)

pect to G(~).

with a s s o c i a t e d m u l t i p l i c a t i v e

R.

is said to satisfy the left Ore condition with res-

tation one derives that
to G(P)

[31].

if for any x • R, g E G(P), there exist x' E R and g ' e G ( P )

such that g'x = x'g.
by ~.

[13],

results one has to impose the left Ore condition on

be a prime ideal of

as before.

and prime ideals of

satisfies the left Ore condition with res-

[21] Proposition 5.5.,

it follows that

R

satisfies

the left Ore condition with respect to G(P) if and only if the elements
of G(P) are units in Qop(R).

This is also equivalent to Qop(P) being

the Jacobson radical of Qop(R); and Qop(R/P)

is then isomorphic to the


www.pdfgrip.com
15

classical

ring of quotients

Moreover,

Op has property

aim of the following
in case

R

Special references

D.C. MURDOCH

is a simple Artinian ring.

The

section is to derive more or less similar results
(prime) ring, with respect to localiza-

T-funetors.

for Section I.

[8]; P. GABRIEL

A.G. HEINICKE

[30],

(T) and Qop(R)

is a left Noetherian

tion at symmetric

V. DLAB

QcI(R/P).

[9]; A.W. GOLDIE

[13]; J. LAMBEK

[19],

[24]; D.C. MURDOCH,

[31]; H. STORRER

[32].

[10],

[11]; O. GOLDMAN

[20]; J. LAMBEK,

F. VAN OYSTAEYEN

G. MICHLER

[26],

[12];
[21];

[2?]; S.K. SIM


www.pdfgrip.com

II.

II.

1. L o c a l i z a t i o n

Unless

at S y m m e t r i c

otherwise

tric T - f u n c t o r .

specified,

The canonical

ring homomorphism.
of j(A)
hand,

Kernel

R

B

is a left
of

THEOREM

For e v e r y

9.
A

PROOF.

of

B

R,

u-torsion,
h a v e that
it f o l l o w s
B = B ce.

that

left i d e a l

= Qo(R)

Qo(R)Cb

Qo(R)j(C)j(x)

j(x) e A e and x e A ee.
j(x) e Q o ( R ) j ( A ) n j(R).

B

and h e n c e

C ' C x c A.

COROLLARY

1.

Qo(1)

Let

Since

On the o t h e r

For e v e r y

left

2), h e n c e

note

Qo(R)/j(R)

By p r o p e r t y
from

is
(T) we

Cb : j(C)b

= Bce,

entailing

first that

Conversely

and then p r o p e r t y

Thus we may w r i t e

j(x)

an ideal

C

= Z' qiai w i t h

in T(o)

and Cx c A + o(R).

!dempotency

(T) y i e l d s

let x e A ec, i.e.,

of o i m p lies

be a left ideal of

R.

such that

Cqi c j(R)

By the left N o e t h e r -

we can find an ideal C' in T(o),

I

Qo(R)j(A)

is said to be the

B ee ; B.

let b e B.

c Qo(R)j(A)

• j(A)

Then Cj(x)
R

is a

If x e A ° then Cx c A for some ideal

for all

for

by A e.

B e = j-I(B)

of Qo(R),

statement

Now choose

ian p r o p e r t y

the e x t e n s i o n

such that Cb c j(R).

qi • Qo (R)' ai • j(A).
i.

then

j : R ~ Qo(R)

= Q o ( R ) b or b e Q o ( R ) ( B n j(R))

A ec = j - I ( Q o ( R ) j ( A ) N j(R)).
Hence

R,

(see P r o p o s i t i o n

To p r o v e the s e c o n d

C e T(o).

of

and o is a symme-

w h e r e A o = {x • R, Cx c A for some C e T(a)}.

is a C e T(o)

Qo(R)j(C)

morphism

w i l l be d e n o t e d

the f i r s t a s s e r t i o n ,

there

Functors.

R.

A ec = A

To p r o v e

A

ideal of Qo(R)

to

FUNCTORS

is left N o e t h e r i a n

R-module

to a left ideal of Qo(R)

if

KERNEL

For a left ideal

contraction

ideal

SYMMETRIC

such that

that C'C • T(o)

C'o(R) = 0
and x • A o.

It is e a s i l y v e r i f i e d

that

= Qo(R)j(1).

COROLLARY

2.

There

is o n e - t o - o n e

correspondence

between maximal

left


www.pdfgrip.com

17

ideals

Proof
that

of Qo(R)

of the

and

last

elements

statement

A e is a m a x i m a l

containing
dicht

A

M c is p r o p e r

in

A

an

and t h e r e f o r e

ideal

mormorphisms
morphism

restricts
if

of

R

is a m a x i m a l

M ce

left

by p r o p e r t y

It is o b v i o u s

a proper

and thus

Consequently
R/K ~ 0

~ Qa(R/K),

follows,

A

since

left

But

then

most

of the

R

then

M c C A for

Thus

M = A e.

but A e is not
sequence

not a l w a y s

yield

if Q~ is exact.

This

following

M,

contra-

of Qa(R)

an e x a c t

does

even

of

ideal

= A e would

ideal

(T).

t h e n A a is an ideal

of Qo(R).

: Q~(R)

in what

M

to

A e C'(a).

M ce c A e or M c A e follows.

0 ~ K ~ R ~-L

Qa~

that

of Qa(R)

and M c ~ T(a)

is an ideal

cessarily

central

R

: Suppose

ideal

Conversely,

some A e C'(o)

If

left

properly,

M # A e.

of C'(o).

results

ne-

of r i n g

ho-

a ring

homo-

problem

is

apply

to s e c t i o n

III.

THEOREM

1.

10.

Let

Qa(T(R))

= ~(Qa(R))

2. The u n i q u e
JT

R-linear

: R ~ QT(R)

induced

PROOF

T ~ a be a r b i t r a r y

1.

The

calization

and

QT(Qa(R))

map

Qo(R)

to Q~(R),

in Qo(R)

and

follows,

sequence

0 ~ ~(R)

an exact

sequence

:

able
and

to p r o o f

equality

some

A,B

ohosen

This

be

entails

that

that

to be

Axc

respective

Axc

ideals

of

T(R/T(R))

the

then

:

canonical

for the r i n g
R-module

~ R ~ R/T(R)

T(Qo(R/T(R)))

while

functors,

structure

structure.

~ 0 yields

under

lo-

~ Qo(R/~(R))

is i m m e d i a t e .

B e T(o)

extending

homomorphism

by t h e i r

exact

Bx = 0 for
may

~ QT(R)

~ Qo(R)

kernel

m Q (R).

is a r i n g

QT(R)

0 ~ Qo(~(R))
If we are

symmetric

Pick

= 0 then

an x e T ( Q o ( R / T ( R ) ) ) .

R/m(R)
R

T(Qo(R)) c Qo(T(R))

we get

for

some A e T(o).

BA c B, h e n c e

= 0 and thus

x = 0.

Then
Since

BA x = 0.

Moreover


www.pdfgrip.com
18

R/o(R)

n T(Qo(R))

: T(R)/o(R)

yields

inclusions

R/~(R) ¢-~ Qo(R)/T(Qo(R))C--~

and t h e r e f o r e

Q~(Qo(R))

give

should

= 0 it f o l l o w s

and so the R - m o d u l e

Q~(R)

a Qo(R)-module

structure

structure,

c o i n c i d e w i t h the s t r u c t u r e

QT(R).

Let JT be the u n i q u e

and let ~,~ be e l e m e n t s

le s t r u c t u r e .

= C~.JT(n)

J

F i n a l l y,

BAh x = 0 w i t h

: C.~JT(n),

linear.

PROPOSITION

11.

Suppose

lows.

that

R.

Hence,

- ~J

in
jy

then

= C JT(~n),
Qo(R)-modu-

(~) • o(QT(R))

a right

that

ideal

= 0,

Qa(~(R))

and a left R-

Bx = 0 for some

R

that

T-functor,

ideal

property

since

pe for some left

~x • T(Qo(R)).

R

C

a left N o e t h e r i a n

o-closed

is a o - p e r f e c t

Now pe is an ideal of Qo(R)

Suppose ABc

of it,

of the

left to prove

T h e n pe is a p r ime

By the left N o e t h e r i a n

for some C e T(o).

We are

if e v e r y p r o p e r

ideal A e of Qo(R).

PROOF.

to

extending

Then J T ( C ~ )

by d e f i n i t i o n

entails

Let o be a s y m m e t r i c

ideal of

uniquely

o-

for some A • T(o).

to a p r o p e r

prime

C~ c R/o(R).

BA • T(T)

R is said to be o - p e r f e c t

closed

map Qo(R) ~ QT(R)

and ~ • Qo(R)

w h i l e A~ c R/o(R)

DEFINITION.

extends

i n d u c e d by ring m u l t i p l i c a t i o n

By 1. it is o b v i o u s l y

If x • T( Q o ( R ) )

B • T(T),

of QT(R)

f o r m this that J ( ~ )

is Qa(R)

is an ideal of Qo(R).
module.

Q (R) is f a i t h f u l l y

w h i c h by the u n i q u e n e s s

R-linear

such that

We d e r i v e

in o t h e r words,

that

of Qo(R).

We may f i n d a C e T(o)
but also J ~ ( C ~ n )

Q~(R)

m Q (R).

2. Since T > o and o ( Q T ( R ) )
injective

:

for

ideal

R

P

extends

be a o-

ideal of Qo(R).

R,

we have that CP
in

we a s s u m e d

ideals A , B of Qo(R).

A C B c c (AB) c c pee = p and t h e r e f o r e

of

ring and let

is not c o n t a i n e d
because

A

ring.

R

P,

o

c p

P = Po fol-

to be o-perfect.

T h e n we have

A c or B c is c o n t a i n e d

in P,

that


www.pdfgrip.com

19

y i e l d i n g that A ce = A or B ee = B is contained in pe.

COROLLARY.

With the above assumptions

:

there is a one-to-one

corres-

pondence between proper prime ideals of Qo(R) and prime ideals of
which are o-closed.
ideals

P

This is easily seen by v e r i f y i n g that proper prime

of Qo(R) restrict to o-closed prime

A,B are ideals of

R

ly A • T(o) yields

such that A B c

Let

1. For every ideal

pC then

ideals of

R.

(AB) e c pCe = p.

Indeed, if
Consequent-

B e c p and B c pC while A ~ T(o) yields AeB e c p,

thus A e or B e is c o n t a i n e d in

P R O P O S I T I O N 12.

R

A

R

P

e n t a i l i n g that

A

be a o-perfect ring, then

of

R,

or

B

is in p C

:

rad A e = (tad A) e

2. There is a one-to-one c o r r e s p o n d e n c e between o-closed left P-primary

ideals of

PROOF.

R

and left pe-primary

ideals of Qo(R).

The previous p r o p o s i t i o n yields that rad A e is intersection of

the extended ideals pe with p e n

A e.

Hence

(rad Ae) e = n {P, p n A and P ~ T(o)}.

If (rad Ae) c c (rad A) ° then (rad A) e = (rad Ae) ce : tad A e will follow.
Therefore,

take x • P for all P n A such that

be an arbitrary prime ideal in T(o),

P

such that P0 D A.

there is an ideal C O • T(o) for which CoX c P0"
minimal prime ideals c o n t a i n i n g

A

is finite,

However,

Then,

if x ~ P0'

Because the n u m b e r of

P

containing

since x (and therefore c e r t a i n l y Cx)

ned in all o-closed minimal prime

Let P0

there exists an ideal

C e T(o) for which Cx c p for every minimal prime
that P e T(o).

is o-closed.

ideals containing

A,

A

such

is contai-

it follows that

Cx C rad A and x e (tad A) o.

2. Recall that an ideal

I

implies B C I or A c rad I.

of

R

is said to be left primary if A B c

Since

R

I

is left N o e t h e r i a n it follows that


www.pdfgrip.com
20

rad I is a prime

P,

and

I

is called a left P-primary

CI ° C I for some ideal C e T(o).

Then P ~ C forces

ideal.

Again,

I o = I and using 1.

the proof becomes easy, following the lines of the proof of P r o p o s i t i o n
11.

Remark.

If

R

is left Noetherian,

closed ideal of

R

and o being idempotent,

is contained in a maximal u-closed

a maximal element in the set of o-closed
ideal.

For, let

A

and

B be ideals of

ideal.

ideals, then
R

P

such that A B c

and B ~ P, then we have that A + P and B + P are in T(o).
(A + P)(B + P) c p contradicts

P ~ T(o).

in the set of o-closed ideals} determines
symmetric,

then T(o)

The set C(o)
o

then every oLet

P

be

is a prime

p with A ~ P
Hence

= {P, P maximal

completely in case o is

is the set of left ideals of

R

c o n t a i n i n g an ideal

which is not contained in any element of C(o).

LEMMA 13.
If

P

Let

R

be an arbitrary ring and let o be a T - f u n c t o r on M(R).

is a left ideal of

dules

X

R

then Qo(P)

in Qo(R) containing P/o(P)

is maximal

in the set of R-submo-

such that X/(P/o(P))

is a u - t o r s i o n

module.

PR00F.

Denote P/a(P) by P.

Qo(Qo(R)/~)

Property

(T) implies that

= Qo(R)/Qo(P)

= Qo(R/P),

and we may derive the following exact sequence

:

0 ~ Qo(P)/~ ~ Qa(R)/p ~ Qo(Qo(R)/~)

The R-module Qo(Qo(R)/p)

~ 0.

is u-torsion free, thus o(Qo(R)/P) c Qo(p)/~,

but since Q a ( p ) / ~ is u-torsion equality follows.
ximal with the desired property.

O b v i o u s l y Qo(P)

is ma-