GP-injective rings need not be P-injective. (English) Zbl 1076.16003

Chen, Jianlong; Zhou, Yiqiang; Zhu, Zhanmin

Commun. Algebra 33, No. 7, 2395-2402 (2005).

Summary: A ring \(R\) is called left P-injective if for every \(a\in R\), \(aR=r(l(a))\) where \(l(\cdot)\) and \(r(\cdot)\) denote left and right annihilators, respectively. The ring \(R\) is called left GP-injective if for any \(0\neq a\in R\), there exists \(n>0\) such that \(a^n\neq 0\) and \(a^nR=r(l(a^n))\). As a response to an open question on GP-injective rings, an example of a left GP-injective ring which is not left P-injective is given. It is also proved here that a ring \(R\) is left FP-injective if and only if every matrix ring \(\mathbb{M}_n(R)\) is left GP-injective.

MSC:

16D50

Injective modules, self-injective associative rings

Keywords:

left FP-injective rings; left GP-injective rings; matrix rings; left P-injective rings

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