Continuous rings with chain conditions
A ring \(R\) is called left continuous if every left ideal of \(R\) is essential in a direct summand of \(R\) and every left ideal of \(R\) isomorphic to a direct summand of \(R\) is itself a direct summand of \(R\). Several interesting characterizations for left continuous rings to be quasi-Frobenius rings are given via the following conditions: \((\text{A}_ 1)\): ACC on left annihilators; \((\text{A}_ 2)\): \(R/\text{Soc} ({_ RR})\) is left Goldie; \((\text{S}_ 1)\): \(S = r(\ell(S))\) for every minimal right ideal \(S\), and \((\text{S}_ 2)\): Every minimal right ideal is essential in a direct summand of \(R_ R\). In fact, it is shown that the quasi-Frobenius rings are the left continuous rings satisfying either \((\text{A}_ 1)\) or \((\text{A}_ 2)\) and either \((\text{S}_ 1)\) or \((\text{S}_ 2)\). These characterizations extend several results [by \textit{P. Ara} and \textit{J. K. Park}, Commun. Algebra 19, 1945-1957 (1991; Zbl 0732.16018), \textit{E. P. Armendariz} and \textit{J. K. Park}, Arch. Math. 58, 24-33 (1992; Zbl 0768.16001), and \textit{V. Camillo} and \textit{M. F. Yousif}, Can. Math. Bull. 34, 462-464 (1991; Zbl 0767.16004)]. Furthermore, the following interesting fact on a certain class of left continuous rings is obtained: Assume that \(R\) is a left continuous ring such that \(R/ \text{Soc} ({_ RR})\) has ACC on left annihilators. If \(S = r(\ell (S))\) for every minimal right ideal \(S\) of \(R\), then \(\text{Soc} ({_ RR}) = \text{Soc} (R_ R)\), \(\text{Soc} (eR)\) is simple for every primitive idempotent \(e\) of \(R\) and there exists a complete set of distinct representatives \(\{Rt_ 1, \dots, Rt_ n\}\) of the isomorphism classes of the simple left \(R\)-modules such that \(\{t_ 1 R, \dots, t_ n R\}\) is a complete set of distinct representatives of the isomorphism classes of the simple right \(R\)-modules.
scientific article; zbMATH DE number 735695
scientific article