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ON THE ENDOMORPHISM RING OF A SEMI-INJECTIVE MODULE

S. Wongwai

Open publisher page 13 citations

Abstract

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 ∈ S\\J(S), there exists a nonzero idempotent α ∈ Ss such that Ker(s) ⊂ Ker(α) and Ker(s(1−α)) 6= 0. The converse is also considered. 1.

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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 ∈ S\\J(S), there exists a nonzero idempotent α ∈ Ss such that Ker(s) ⊂ Ker(α) and Ker(s(1−α)) 6= 0. The converse is also considered. 1.

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Available abstract

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 ∈ S\\J(S), there exists a nonzero idempotent α ∈ Ss such that Ker(s) ⊂ Ker(α) and Ker(s(1−α)) 6= 0. The converse is also considered. 1.

Key concepts: Mathematics, Injective function, Injective module, Endomorphism, Endomorphism ring, Module, Simple module, Divisible group

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