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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 aa
Department of Mathematics, Pondicherry University, Pondicherry, IndiaTo cite this Article
Jothilingam, P. and Mangayarcarassy, S.(1993) 'A note on flatness and injectivity of simple modulesover a commutative ring', Communications in Algebra, 21: 2, 675 — 678
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COMMUNICATIONS
IN ALGEBRA, 21(2), 675-678 (1993)A
NOTE ON FLATNESS AND INJECTIVITY OFSIMPLE MODULES
OVER A COMMUTATIVE RINGP. Jothilingam and
S. Mangayarcarassy
Department of Mathematics
Pondicherry University
Pondicherry
- 605 014India
The main theorem of the paper "Flatness and Injectivity
of Simple Modules over a Commutative Ring" by
Xu Jinzhongwhich appeared in Communications in Algebra
19(2), 535-537(1991)
asserts that a simple unital module over a commutativeassociative 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
beunital.
Theorem
: Let R- A be a homomorphism of rings. Assume A is anoetherian 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 ofA-modules. By flatness, the functor
- BR A : '$38 is exact. Theself injectivity of A implies that the functor HomA (-,A) @-@is
exact. Hence the composite functor HornA
( - a R ~ , ~:)e+ 2lis exact.But there is a natural isomorphism of functors Homh
(-@ A,A) RHorn2(-,A) from the category
e to the category 3 . So Hom (-,A) Ris exact on the category
. This precisely means that A isR-injective.
We next prove
+ .To show A is R-flat, it is enough to show that whenever
O-rM'+
M 'M" 7 0is an exact sequence of finitely generated R-modules, then the
tensored sequence
O;-* M'
eR A-tM QRA-vM1' SRA+Ois exact. Now being R-injective, the sequence
0-Hom (M" ,A)+HomR(M,A)3HomR(M' ,A)4O
R
is exact. Using the isomorphism HomA
(-% A,A) HornR (-,A), weget an exact sequence of A-modules
:Using
* to denote A-duls, the sequence above can be written as0
-+ (M"IR A)*+ (M L¶RA)*--(M+I I R ~ ) * 30which is exact. Taking A-duals in this exact sequence, and using
the fact that A is self injective we get the exact sequence
**
* *o
--c (MI B ~ A ) -+ (M I ~ A ) 3 (M" B~A)**+ o . . . (2)-
Claim : Every finitely generated A-module L is reflexive.For, take a finite presentation
Downloaded At: 12:05 19 January 2010
FLATNESS AND INJECTIVITY OF SIMPLE MODULES 677
which exists since A is a noetherian ring; here
Fa and F1; arefinitely generated free A-modules. A being self injective,
taking A-duals preserves exactness of a sequence. Hence
F*'
--+ F** -+ L*- 0 is exact. The commutative diagram below1
0and natural isomorphisms
f and gimply that the natural homomorphism L
-b L** is an isomorphism,i.e.,
L is reflexive. Hence the claim.Now M' BRA, M %A and M" %A are all finitely generated
A-modules. Hence by what precedes, they are naturally isomorphic
to their double A-duals. Using this in (21, we see that the
sequence (1) is exact. Hence A is R-flat.
Remark
1 : Taking A = R/m, where m is a maximal ideal of R, weget the main result of Xu Jinzhong.
Remark 2
: For the part A is R-flat =$ A is R-inj.ective, wedon't need the assumption that
A must be a noetherian selfinjective ring.
The referee has suggested the following generalization
of the above Theorem.
If
A is a noetherian, self-injective ring then it isquasi-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:
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678 JOTHILINGAM AND MANGAYARCARASSY
Let R
-3 A be a ring homomorphism. If A is an injectivecogenerator in the category of A-modules then
A is R-flat if andonly 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 inAdvanced Mathematics, No.8, 1986.
3. Xu Jindong, "Flatness and Injectivity
of Simple Modulesover
a Commutative Ring" Communications in Algebra,-
19(2), 1991, 535-537.Received:
December 1991Revised:
April 1992Downloaded At: 12:05 19 January 2010
