On admissible rings
Let \(R\) be an associative ring with an identity, \(R^+\) the additive group of \(R\), \(\mathbb{C}\) the field of complex numbers, \(\Omega=\text{Char }R=\text{Hom}_Z(R^+,\mathbb{C}^*)\). The group \(\Omega\) is an \(R\)-\(R\)-bimodule with operators defined by \((a\chi b)(c)=\chi(bca)\), \(\chi\in\Omega\), \(a,b,c\in R\). A finite ring \(R\) is called left (right) admissible if there exists \(\chi\in\Omega\) such that \(\Omega=R\chi\) (\(\Omega=\chi R\)). This notion was introduced by \textit{H. L. Claasen} and \textit{R. W. Goldbach} [Indag. Math., New Ser. 3, No. 1, 11-26 (1992; Zbl 0771.16009)] who posed the question whether a finite left admissible ring is right admissible. As a special case of this question they also asked whether there exists a finite local ring with one minimal left and more than one minimal right ideal. Answering the first question the author proves that for a finite ring \(R\) the following statements are equivalent: (i) \(R\) is left admissible; (ii) \(R\) is right admissible; (iii) \(R\) is a Frobenius ring. Let, further, \(K(x)\) be the rational function field over a field \(K\). The author constructs an example of an artinian infinite local ring \(R\subset M_2(K(x))\) with one minimal right ideal and more than one minimal left ideal and proves that any right artinian local ring with an identity finitely generated as a module over its center has one minimal right ideal if and only if it has one minimal left ideal.
scientific article; zbMATH DE number 1029285
scientific article