A field-like property of finite rings
Let \(R\) be a finite ring and \(R^ +\) its additive group. The group \(\Omega\) of characters of \(R^ +\) is an \(R\)-bimodule with operators defined by \((a\omega b)(x) = \omega(bxa)\), \(\omega \in \Omega\), \(a,b,x \in R\). The authors call the ring \(R\) left (right) admissible if \(\Omega\) is isomorphic to \(R^ +\) as a right (left) \(R\)-module. The admissibility of finite rings, called their field-like property in the paper, is investigated for rings with a few minimal one-sided ideals, for semi- simple, local and commutative rings. In particular, the relation between the admissibility and the structure of the one-sided socle of \(R\) is considered and general necessary and sufficient conditions of the admissibility are established.
scientific article; zbMATH DE number 57228