On simple GP-injective modules. (English) Zbl 0840.16006

Nam, S. B.; Kim, N. K.; Kim, J. Y.

Commun. Algebra 23, No. 14, 5437-5444 (1995).

A right \(R\)-module \(M\) is called general right principally injective (briefly right GP-injective) if for every \(0\neq a\in R\) there exists a positive integer \(n\) such that \(a^n\neq 0\) and every \(R\)-homomorphism from \(a^nR\) to \(M\) can be extended to one from \(R\) to \(M\). A ring \(R\) is said to be right GP-injective if the right \(R\)-module \(R_R\) is GP-injective. A ring \(R\) is called right quasi-duo if every maximal right ideal is a two-sided ideal.
It is shown that if \(R\) is a right GP-injective ring, then the Jacobson radical of \(R\) is equal to the right singular ideal of \(R\). Also it is proved that if \(R\) is a right quasi-duo ring, then every simple right \(R\)-module is GP-injective if and only if \(R\) is a von Neumann regular ring if and only if \(R\) is a \(V\)-ring.

Reviewer: J.K.Park (Pusan)

MSC:

16D50	Injective modules, self-injective associative rings
16D60	Simple and semisimple modules, primitive rings and ideals in associative algebras
16E50	von Neumann regular rings and generalizations (associative algebraic aspects)

Keywords:

general right principally injective modules; right GP-injective rings; simple right modules; right quasi-duo rings; von Neumann regular rings

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