Generalizations of theorems of Fuller
Let \(R\) be a right Artinian ring and \(e\) a primitive idempotent of \(R\). K. Fuller showed that [\textit{F. W. Anderson} and \textit{K. R. Fuller}, Rings and Categories of Modules, 2nd ed. (1992; Zbl 0765.16001); Theorem 31.3]\ the following conditions are equivalent: (1) \(eR\) is an injective right \(R\)-module. (2) There exists a primitive idempotent \(f\) of \(R\) such that (2*), \(S(eR)\cong T(fR)\) and \(S(Rf)\cong T(Re)\), where \(S(M)\) and \(T(M)\) denote the socle and the top of \(M\), respectively. (3) There exists a primitive idempotent \(f\) of \(R\) such that (3l), \(\ell_{eR}(r_{Rf}(I))=eI\) for each left ideal \(I\), and (3r), \(r_{Rf}(\ell_{eR}(K))=Kf\) for each right ideal \(K\) of \(R\). In this paper \(R\) is a semiprimary ring. For a right \(R\)-module \(M\) with \(S(M)\cong T(fR)\) and \(P=\text{End }M\), the authors consider a pair \((_PM,Rf_{fRf})\) instead of an \(i\)-pair \((_{eRe}eR,Rf_{fRf})\), and give generalizations of the theorem of Fuller. The results obtained in this paper are useful in ring theory.
scientific article; zbMATH DE number 1117481