Regular gamma rings (Q5903628)

Let \((\Gamma,M)\) be a \(\Gamma\)-ring, where \(\Gamma,M\) are both additive groups. \((\Gamma,M)\) is regular if for each \(a\in M\), there exists \(\delta\in\Gamma\) such that \(a\delta a=a\). In this paper, some familiar results on regular rings are generalized to regular \(\Gamma\)-rings. For example, if \((\Gamma,M)\) is semiprime with max-\(\gamma\) and min-\(\ell\), then every left \(L\)-module and every left \(R\)-module are regular. Relations among regularity of the operator rings \(L\), \(R\) and \((\Gamma,M)\) are also investigated. In particular, if left and right unities exist in \((\Gamma,M)\), then \(L\) (or \(R\)) is regular iff \(M\) is regular. As an extension of this result, the regularity of a Morita context is considered. The authors point out that regularity is in fact one of the Morita invariants.