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**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.

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.

Reviewer: Frieda ThĂ©ron (Pretoria)

### MSC:

16D50 | Injective modules, self-injective associative rings |

16E50 | von Neumann regular rings and generalizations (associative algebraic aspects) |