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Zhou, Yiqiang

Rings in which certain right ideals are direct summands of annihilators. (English) Zbl 1020.16003

J. Aust. Math. Soc. 73, No. 3, 335-346 (2002).
A ring \(R\) is called left AGP-injective when, for any \(0\neq a\in R\), there exists \(n>0\) such that \(a^n\neq 0\) and \(a^nR\) is a direct summand of a right annihilator ideal (these rings generalize left principally injective rings, which are the rings such that each principal right ideal is a right annihilator). The author studies properties of these rings and, among other results, shows that every left AGP-injective ring with ACC on left annihilators is semiprimary. Moreover, if \(R\) is a right Noetherian left AGP-injective ring such that every right ideal of \(R\) which is isomorphic to \(R_R\) is a direct summand of \(R_R\), then \(R\) is right Artinian.
Reviewer: J.L.Gómez-Pardo (Santiago de Compostela)

Cited in 1 Review
Cited in 10 Documents

MSC:

16D50 Injective modules, self-injective associative rings
16P60 Chain conditions on annihilators and summands: Goldie-type conditions

Keywords:

AP-injective rings; AGP-injective rings; GC2 condition; quasi-dual rings; direct summands; annihilator ideals
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\textit{Y. Zhou}, J. Aust. Math. Soc. 73, No. 3, 335--346 (2002; Zbl 1020.16003)
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