Principally Small Injective Rings
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A right ideal I of a ring R is small in case for every proper right ideal K of R, K ＋ I ≠ R. A right R-module M is called PS-injective if every R-homomorphism f : aR → M for every principally small right ideal aR can be extended to R → M. A ring R is called right PS-injective if R is PS-injective as a right R-module. We develop, in this article, PS-injectivity as a generalization of P-injectivity and small injectivity. Many characterizations of right PS-injective rings are studied. In light of these facts, we get several new properties of a right GPF ring and a semiprimitive ring in terms of right PS-injectivity. Related examples are given as well.
Abstract
A right ideal I of a ring R is small in case for every proper right ideal K of R, K ＋ I ≠ R. A right R-module M is called PS-injective if every R-homomorphism f : aR → M for every principally small right ideal aR can be extended to R → M. A ring R is called right PS-injective if R is PS-injective as a right R-module. We develop, in this article, PS-injectivity as a generalization of P-injectivity and small injectivity. Many characterizations of right PS-injective rings are studied. In light of these facts, we get several new properties of a right GPF ring and a semiprimitive ring in terms of right PS-injectivity. Related examples are given as well.
참고문헌 (12)
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