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In a one-dimensional local ring R with finite integral closure each nonzerodivisor has a value in N-d, where d is the number of maximal ideals in the integral closure. The set of values constitutes a semigroup, the value semigroup of R. We investigate the connection between the value semigroup and the ring. There is a particularly close connection for some classes of rings, e.g. Gorenstein rings, Arf rings, and rings of small multiplicity. In many respects, the Arf rings and the Gorenstein rings turn out to be opposite extremes. We give applications to evenings, intersection numbers, and multiplicity sequences in the blow-up sequences studied by Lipman. (C) 2000 Elsevier Science B.V. All rights reserved

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For some numerical semigroup rings of small embedding dimension, namely those of embedding dimension 3, and symmetric or pseudosymmetric of embedding dimension 4, presentations has been determined in the literature. We extend these results by giving the whole graded minimal free resolutions explicitly. Then we use these resolutions to determine some invariants of the semigroups and certain interesting relations among them. Finally, we determine semigroups of small embedding dimensions which have strongly indispensable resolutions. (C) 2013 Elsevier B.V. All rights reserved

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A family of quotient rings of the Rees algebra associated to a commutative ring is studied. This family generalizes both the classical concept of idealization by Nagata and a more recent concept, the amalgamated duplication of a ring. It is shown that several properties of the rings of this family do not depend on the particular member.MSC: 20M14; 13H10; 13A30.

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