definition of injective modules

46 2 Definition An R-module J is an injective module if J satisfies one of the equivalent conditions of Proposition 46 1 1) J is an injective module 2) For every left ideal I < R and for every homomorphisms of R-modules f : I → J there is a homomorphism ¯ f : R → J such ¯ fI = f

A ring R is Noetherian if and only if every direct sum of injective R-modules is injective Proof We first show that if M is a finitely generated R-module, then HomR( 

27 nov 2009 · M Lemma 1 2 Let M be a module Then M is injective iff HomR(−,M) is exact Proof Let 0 → N → N 

We will thus require that I be closed under retracts and products so that I = I and we define: Definition 1 2 A collection I of R-modules is an injective class if it is 

Definition 2 Let A and B be modules over a ring R A function f : A −→ B is an R- module homomorphism provided that for 

Then we use a “change of rings” functor to prove the assertion in general Definition 1 (10 6 5 in Ash) Let R be an integral domain A R-module M is called divisible 

Let M =E(K)J thus X = 0 so that E(K) is subisomorphic to K From the proof of Theorem 1 1, one can see that in the definition of a ker- injective module ker hI:; 

injective and quasi-injective left modules over all arbitrary ring It with identity r E R It is easy to check that y' is a well-defined homomorphism, so (Co , yo) ( C' 

By means of this condition we prove a structure theorem (Corollary 2 3) for rings having both a left and a right injective projective irreducible module with the same