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Communications in Algebra

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A note on flatness and injectivity of simple modules over a commutative

ring

P. Jothilingam a; S. Mangayarcarassy a

a Department of Mathematics, Pondicherry University, Pondicherry, India

To cite this Article Jothilingam, P. and Mangayarcarassy, S.(1993) 'A note on flatness and injectivity of simple modules

over a commutative ring', Communications in Algebra, 21: 2, 675 — 678

To link to this Article: DOI: 10.1080/00927879308824588

URL: http://dx.doi.org/10.1080/00927879308824588

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COMMUNICATIONS IN ALGEBRA, 21(2), 675-678 (1993)

A NOTE ON FLATNESS AND INJECTIVITY OF

SIMPLE MODULES OVER A COMMUTATIVE RING

P. Jothilingam and

S. Mangayarcarassy

Department of Mathematics

Pondicherry University

Pondicherry - 605 014

India

The main theorem of the paper "Flatness and Injectivity

of Simple Modules over a Commutative Ring" by Xu Jinzhong

which appeared in Communications in Algebra 19(2), 535-537

(1991) asserts that a simple unital module over a commutative

associative ring with identity is flat if and only if it is

injective. In this note we establish a more general result

by very elementary means.

Throughout we will consider only commutative

associative rings with identity and all our .modules will be

unital.

Theorem : Let R- A be a homomorphism of rings. Assume A is a

noetherian ring and self injective. Then A is R-flat if and only

Copyright Q 1993 by Marcel Dekker, Inc.

Downloaded At: 12:05 19 January 2010

676 JOTHILlNGAM AND MANGAYARCARASSY

-Proof : We prove *.

Let be the category of R-modules and 3 the category of

A-modules. By flatness, the functor - BR A : '$38 is exact. The

self injectivity of A implies that the functor HomA (-,A) @-@is

exact. Hence the composite functor HornA ( - a R ~ , ~:)e+ 2lis exact.

But there is a natural isomorphism of functors Homh (-@ A,A) R

Horn2(-,A) from the category e to the category 3 . So Hom (-,A) R

is exact on the category . This precisely means that A is

R-injective.

We next prove + .

To show A is R-flat, it is enough to show that whenever

O-rM'+ M 'M" 7 0

is an exact sequence of finitely generated R-modules, then the

tensored sequence

O;-* M' eR A-tM QRA-vM1' SRA+O

is exact. Now being R-injective, the sequence

0-Hom (M" ,A)+HomR(M,A)3HomR(M' ,A)4O

R

is exact. Using the isomorphism HomA (-% A,A) HornR (-,A), we

get an exact sequence of A-modules :

Using * to denote A-duls, the sequence above can be written as

0 -+ (M"IR A)*+ (M L¶RA)*--(M+I I R ~ ) * 30

which is exact. Taking A-duals in this exact sequence, and using

the fact that A is self injective we get the exact sequence

** * *

o --c (MI B ~ A ) -+ (M I ~ A ) 3 (M" B~A)**+ o . . . (2)

-Claim : Every finitely generated A-module L is reflexive.

For, take a finite presentation

Downloaded At: 12:05 19 January 2010

FLATNESS AND INJECTIVITY OF SIMPLE MODULES 677

which exists since A is a noetherian ring; here Fa and F1; are

finitely generated free A-modules. A being self injective,

taking A-duals preserves exactness of a sequence. Hence

F*' --+ F** -+ L*- 0 is exact. The commutative diagram below

1 0

and natural isomorphisms f and g

imply that the natural homomorphism L -b L** is an isomorphism,

i.e., L is reflexive. Hence the claim.

Now M' BRA, M %A and M" %A are all finitely generated

A-modules. Hence by what precedes, they are naturally isomorphic

to their double A-duals. Using this in (21, we see that the

sequence (1) is exact. Hence A is R-flat.

Remark 1 : Taking A = R/m, where m is a maximal ideal of R, we

get the main result of Xu Jinzhong.

Remark 2 : For the part A is R-flat =$ A is R-inj.ective, we

don't need the assumption that A must be a noetherian self

injective ring.

The referee has suggested the following generalization

of the above Theorem.

If A is a noetherian, self-injective ring then it is

quasi-frobenius. Hence it is an injective cogenerator of the

category of A-modules. This means

0 4 M ' 4 M 4 M " 4 0

is an exact sequence of A-modules if and only if.

0 4 Hom (Mtt,A)4 Hom (M,A) 4 Horn (M',A)

is exact. (This can be used to show the exactness of (1)).

Hence the Theorem can be stated as follows:

Downloaded At: 12:05 19 January 2010

678 JOTHILINGAM AND MANGAYARCARASSY

Let R -3 A be a ring homomorphism. If A is an injective

cogenerator in the category of A-modules then A is R-flat if and

only if A is R-injective.

ACKNOWLEDGEMENT

The second author was financially supported by National

Board for Higher Mathematics, India,

REFERENCES

1. Atiyah and Macdonald, An introduction to Commutative

Algebra, Addison-Wesley Publishing Company, 1969.

2 . Matsumara, Commutative Ring Theory, Cambridge Studies in

Advanced Mathematics, No.8, 1986.

3. Xu Jindong, "Flatness and Injectivity of Simple Modules

over a Commutative Ring" Communications in Algebra,

-19(2), 1991, 535-537.

Received: December 1991

Revised: April 1992

Downloaded At: 12:05 19 January 2010

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ادامه مطلب
+ نوشته شده در  دوشنبه 1388/08/18ساعت 16:22  توسط محمد صادق غفارزاده نمازی  | 

تاریخچه ایگوروف

http://www.gap-system.org/~history/Biographies/Egorov.html

Dimitri Fedorovich Egorov


Born: 22 Dec 1869 in Moscow, Russia
Died: 10 Sept 1931 in Kazan, USSR

Dimitri Egorov attended school in Moscow then entered Moscow University to study mathematics and physics, enrolling in 1887. The teacher to influence him most at this time was Bugaev. Egorov wrote his first paper in 1892 on numerical integrals and derivatives clearly influenced by Bugaev's work in this area.

Egorov taught at Moscow University from 1894, obtaining a doctorate in 1901. He spent a year abroad, then in 1903 he returned to become a professor at Moscow University.

Egorov worked on triply orthogonal systems and potential surfaces, making a major contribution to differential geometry. Some of Egorov's work was presented by Darboux in his famous four volume work Leçons sur la théorie général des surfaces et les applications géométriques du calcul infinitésimal.

Egorov also worked on integral equations and a theorem in the theory of functions of a real variable is named after him. Luzin was Egorov's first student and became a member of the school Egorov created in Moscow dealing with functions of a real variable.

In 1917 Egorov became secretary of the Moscow Mathematical Society. Then in 1921 he was elected vice-president, becoming president the following year. In 1923 Egorov became director of the Institute for Mechanics and Mathematics at Moscow State University.

However Egorov was a deeply religious man and when the Church was repressed after the revolution, Egorov defended them. In 1922-23 there were mass execution of clergy and in 1928 the attack was renewed. Egorov was in a position of power in the Moscow Mathematical Society and he tried to shelter academics who had been dismissed from their posts. He tried to prevent the attempt to impose Marxist methodology on scientists.

In 1929 Egorov was dismissed as director of the Institute for Mechanics and Mathematics and given a public rebuke.

Some time later he was arrested as a "religious sectarian" and put in prison. The Moscow Mathematical Society continued to support Egorov, refusing to expel him, and those who presented papers at the next meeting, including Kurosh, were to be expelled by an "Initiative group" who took over the Society in November 1930. They expelled Egorov denouncing him as

a reactionary and a churchman.

Egorov went on a hunger strike in prison and eventually, by this time close to death, he was taken to the prison hospital in Kazan. Chebotaryov's wife was working as a doctor in the prison hospital and, although it sounds rather unlikely, it is reported that Egorov died at Chebotaryov's home.

 

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