ACTA MATHEMATICA UNIVERSITATIS COMENIANAE

Vol. LXXI, 1(2002)
p. 27

ON THE ENDOMORPHISM RING OF A SEMI-INJECTIVE MODULE
S. Wongwai

Abstract.  Let $R$ be a ring. A right $R$-module $M$ is called quasi-principally (or semi- ) injective if it is $M$-principally injective. In this paper, we show: (1) The following are equivalent for a projective module $M$: (a) Every $M$-cyclic submodule of $M$ is projective; (b) Every factor module of an $M$-principally injective module is $M$-principally injective; (c) Every factor module of an injective $R$-module is $M$-principally injective. (2) The endomorphism ring $S$ of a semi-injective module is regular if and only if the kernel of every endomorphism is a direct summand. (3) For a semi-injective module $M$, if $S$ is semiregular, then for every $s\in S\setminus J(S),$ there exists a nonzero idempotent $\alpha\in Ss$ such that $\ker(s)\subset\ker(\alpha)$ and $\ker(s(1-\alpha))\not = 0.$ The converse is also considered.

AMS subject classification:  16D50, 16D70, 16D80
Keywords:  Semi-injective modules, Endomorphism rings