On QF-rings with cyclic Nakayama permutations
Let \(R\) be a basic quasi-Frobenius ring and \(P(R)=\{P_1,\dots,P_n\}\) a maximal system of non-isomorphic indecomposable projective \(R\)-modules. The Nakayama-permutation \(\pi\in S_n\) of \(R\) is defined by \(\text{Soc }P_i\simeq P_{\pi(i)}/\text{rad }P_{\pi(i)}\), \(i=1,\dots,n\). Main result of the paper is the following: Let \(R\) have the property that for every idempotent \(e\in R\): (i) The ring \(eRe\) is a quasi-Frobenius ring, and (ii) The Nakayama-permutation of \(eRe\) permutes the projective \(eRe\)-modules in \(P(eRe)\) cyclically. Then there exists a local quasi-Frobenius ring \(S\), an element \(c\in\text{rad }S\) and an automorphism \(\sigma\) of \(S\) satisfying \(\sigma(c)=c\), \(\sigma(x)c=cx\) \(\forall x\in S\), such that \(R\) is isomorphic to the generalized matrix ring \(A_n(S,\sigma,c)\) which, in case of Nakayama-rings, was introduced [in Arch. Math. 26, 23-35 (1975; Zbl 0304.16006)] by the reviewer.
scientific article; zbMATH DE number 1014602