Modules injective with respect to maximal ideals
by
Septimiu Crivei
"Babes-Bolyai" University of Cluj-Napoca

We denote by R an associative ring with non-zero identity and all modules are left unital R-modules. If A is a module, then we denote by Soc(A) the socle of A and by E(A) the injective envelope of A. A module A is said to be semiartinian if every non-zero homomorphic image of A contains a simple submodule. A module A is said to have the property (P) if Soc(A)=0 and every non-zero proper factor module of A has non-zero socle.

We introduce the notion of m-injectivity. A module D is said to be m-injective if any homomorphism from any maximal left ideal of R to D extends to R. It is shown that for a semiartinian ring R, every m-injective R-module is injective. There are given examples of m-injective modules which are not injective.

We prove that for a module D the following statements are equivalent : (i) D is m-injective ; (ii) D is injective with respect to every short exact sequence of modules 0 --> A --> B --> C --> 0 where C is a non-zero semiartinian module.

It is defined the notion of m-injective envelope of a module A, denoted by E_m(A). Then every module A has an m-injective envelope E_m(A) contained in E(A), unique up to an isomorphism.

We shall introduce the notion of minimal m-injective module, which generalizes the notion of indecomposable injective module. A non-zero m-injective module D is said to be a minimal m-injective module if D is an m-injective envelope of each of its non-zero submodules. It will be established the structure of a minimal m-injective module as an m-injective envelope of a module which is either simple or has the property (P).

References

[1] Crivei, I., On a class of modules closed to non-zero submodules, Automat. Comput. Appl. Math., vol. 1, No. 2, (1992), 94-98.

[2] Crivei, S., m-injective modules, Mathematica (Cluj), 40 (63), No.1 (1998), 71-78.

[3] Crivei, S., Minimal m-injective modules, Mathematica (Cluj), 40 (63), No.2 (1998), 159-164.

[4] Johnson, J.L., Modules injective with respect to primes, Comm.Alg. 7, no.3, (1979), 327-332.

[5] Nastasescu, C., Inele. Module. Categorii, Ed. Academiei, Bucuresti, 1976.

[6] Sharpe, D.W., Vámos, P., Injective modules, Cambridge Univ. Press, 1972.

Date received: May 29, 2000