On Artinian QF-3, 1-Gorenstein rings (Q1088763)

A Noetherian ring R is called 1-Gorenstein, if \(id(_ RR)\leq 1\) and \(id(R_ R)\leq 1\), where id denotes the injective dimension. Extending results of \textit{M. Harada} [Osaka J. Math. 2, 357-368 (1965; Zbl 0166.304)] and \textit{Y. Iwanaga} [ibid. 15, 33-46 (1978; Zbl 0402.16017)], the author proves the following Theorem I: Let R be a connected Artinian ring, which is not quasi-Frobenius. Then the following conditions are equivalent: (1) R is a hereditary QF-3 ring, (2) R is Morita equivalent to a triangular matrix ring over a division ring, (3) R is a left and right serial 1-Gorenstein ring, (4) (resp. (5)) R is a QF-3 1-Gorenstein ring with a simple projective (resp. injective) left module. The second main result is Theorem II: Let R be an Artinian QF-3 1- Gorenstein ring and P an indecomposable projective R-module. If P is distributive (in the sense of \textit{V. Camillo} [J. Algebra 36, 16-25 (1975; Zbl 0308.16015)], then the socle of P has length \(\leq 2\). The last section contains several examples in the form of quiver algebras. E.g. it is shown, that distributivity is necessary in Theorem II and that for any m, \(2\leq m\leq \infty\), there is a QF-3 1-Gorenstein algebra, whose maximal quotient ring Q satisfies \(id(_ QQ)=id(Q_ Q)=gl.\dim. Q=m\).