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Gorenstein rings with semigroup bases. (English) Zbl 0549.16007
The author develops the module theory over non-semiprime QF-3, 1- Gorenstein rings with zero socle. First the author proves that if R is a discrete valuation ring and S is a semi-Kupisch semigroup which is Frobenius with respect to R, then the semigroup ring R[S] is a QF-3, 1- Gorenstein ring with zero socle. In order to study the module theory, the author introduces the notions of ring frame and frame ring over a semi- Kupisch semigroup. Then he proves that if \(\Lambda\) is a QF-3, 1- Gorenstein ring with zero socle, then any injective indecomposable \(\Lambda\) -module E is either torsionfree or torsion with respect to the Lambek torsion theory. And he also investigates finitely generated torsion modules over a Kupisch ring and finitely generated torsionfree modules over a strongly Kupisch ring.
Reviewer: Y.Xu
MSC:
16L60 Quasi-Frobenius rings
20M25 Semigroup rings, multiplicative semigroups of rings
16P40 Noetherian rings and modules (associative rings and algebras)
16P60 Chain conditions on annihilators and summands: Goldie-type conditions
16D50 Injective modules, self-injective associative rings
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
16S90 Torsion theories; radicals on module categories (associative algebraic aspects)
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