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On Some Characteristic Properties of Self-Injective Rings

Kwangil Koh

1968
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Proceedings of the American Mathematical Society
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A ring with unit element is said to be left self-infective if and only if every (left) 2?-homomorphism of a left ideal of R into R can be given by the right multiplication of an element of R. In [l], Ikeda-Nakayama introduced the following conditions in a ring R with unit element: (A) Every (left) 2?-homomorphism of a principal left ideal of R into R may be given by the right multiplication of an element of R. (Ao) Every (left) 2?-homomorphism of a principal left ideal L of R into a residue
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... le R/L', of R modulo a left ideal L', maybe obtained by the right multiplication of an element, say c, of R: x->xc (mod L'), ixEL). (B) If I is a finitely generated right ideal in 2?, then the set of right annihilators of the set of left annihilators of I is 2. (B*) If 2 is a principal right ideal in R, then the set of right annihilators of the set of left annihilators of 2 is I. We introduce another condition: (C) If F is a finitely generated left free 2?-module and M is a cyclic submodule of F then any 2?-homomorphism of M into R can be extended to a 2?-homomorphism of F into R. In this paper, we shall prove the following: In a ring with 1, (B) holds if and only if (C) holds. If 2? is a ring with 1 such that every principal left ideal is projective, then the three conditions (A), (A0) and (B) are equivalent. If 2? is a ring with 1 such that the right singular ideal (refer to [4] for definition) is zero, then 2? is a semisimple ring with minimum conditions on one-sided ideals if and only if R satisfies the maximum condition for annihilator right ideals and the condition (B). In particular, a regular ring R with 1 is a semisimple ring with minimum conditions on one-sided ideals if and only if it satisfies the maximum condition for annihilator right ideals. In a simple ring R with 1, the condition (B*) and the existence of a maximal annihilator left ideal in R are necessary and sufficient conditions for R to satisfy minimum conditions on one-sided ideals. In a ring with 1, the condition (B*) implies that the left singular ideal of R is, indeed, the Jacobson radical of R. In the sequel, if A is a subset in R, we denote the set of left (right)

doi:10.2307/2036173
fatcat:3cptebupgbbylmunny5jftgs3q