## Generalizations of principal injectivity.(English)Zbl 0949.16002

A left $$R$$-module $$M$$ is called principally injective if for any principal left ideal $$P$$ of $$R$$, every $$R$$-homomorphism of $$P$$ into $$M$$ extends to one from $$R$$ into $$M$$. A left $$R$$-module $$M$$ is called GP-injective if for any $$0\neq a\in R$$, there exists a positive integer $$n$$ (depending on $$a$$) such that any $$R$$-homomorphism of $$Ra^n$$ into $$M$$ extends to one of $$R$$ into $$M$$. YJ-injectivity is defined exactly as GP-injectivity except for the additional requirement that the positive integer $$n$$ must also be such that $$a^n\neq 0$$.
These concepts have been studied before, but new characterizations are found here. An element $$a$$ of $$R$$ is said to be $$\pi$$-regular if there exists a positive integer $$m$$ such that $$a^m=a^mba^m$$ for some $$b\in R$$. A subset $$U$$ of $$R$$ is called $$\pi$$-regular if every element in $$U$$ is $$\pi$$-regular. $$\pi$$-regularity is characterized in terms of GP-injectivity in various ways, of which the most important is probably the one stating that $$R$$ is $$\pi$$-regular if and only if every left $$R$$-module is GP-injective. These characterizations pave the way for new equivalences of von Neumann regular rings and it is proved inter alia that $$R$$ is von Neumann regular if and only if every left $$R$$-module is YJ-injective. Thus two questions posed by R. Yue Chi Ming [Riv. Mat. Univ. Parma, IV. Ser. 11, 101-109 (1985; Zbl 0611.16011); ibid. 13, 19-27 (1987; Zbl 0682.16009); ibid., V. Ser. 5, 183-188 (1996; Zbl 0877.16002)] are answered in the affirmative.

### MSC:

 16D50 Injective modules, self-injective associative rings 16E50 von Neumann regular rings and generalizations (associative algebraic aspects)

### Citations:

Zbl 0611.16011; Zbl 0682.16009; Zbl 0877.16002