Let Q be a parameter ideal of a Noetherian local ring (R, m). The Goto number g(Q) of Q is the largest integer g such that Q : m is integral over Q. We examine the values of g(Q) as Q varies over the parameter ideals of R. We concentrate mainly on the case where dimR = 1, and many of our results concern parameter ideals of a numerical semigroup ring.

Abstract We uncover a connection between the model-theoretic notion of superstability and that noetherian rings pure-semisimple rings. characterize via class left modules with embeddings. Theorem 0.1 For ring R following are equivalent. (1) is noetherian. (2) The R-modules embeddings superstable. (3) every ? ? | + ? 0 , there ? such has uniqueness limit models cardinality ?. (4) Every model in ...

Let R be a commutative Noetherian local ring. This paper deals with the problem asking whether R is Gorenstein if the nth syzygy module of the residue class field of R has a non-trivial direct summand of finite Gdimension for some n. It is proved that if n is at most two then it is true, and moreover, the structure of the ring R is determined essentially uniquely.

Let R be a commutative ring with identity. In this paper we define the notion of almost primary ideal. An ideal of a ring R is said to be an almost primary ideal if for a, b ∈ R with ab ∈ I − I2 then a ∈ I or bn ∈ I for some n ∈ N. We prove that in Noetherian domains almost primary ideals are primary. Mathematics Subject Classification: 13A15

We prove that the elements of any standard basis of In, where I is an ideal of a Noetherian local ring and n is a positive integer, have order bounded by a linear function in n. We deduce from this that the elements of any standard basis of In in the sense of Grauert-Hironaka, where I is an ideal of the ring of power series, have order bounded by a polynomial function in n. The aim of this pape...

Let (R, m) be a Noetherian local ring and let I be an m-primary ideal. In this paper we give sharp bounds on the multiplicity of the special fiber ring F of I in terms of other well-known invariants of I . A special attention is then paid in studying when equality holds in these bounds, with a particular interest in the unmixedness or, better, the Cohen-Macaulayness of F .

Meta-projective modules on a ring R were essentially studied by Feng Lianggui and Tong Wenting in [3] when R is a commutative ring. While Duo-rings were studied by A. L. Fall in [2] and by M. Sanghare in [8]. The results of [3] were extended by M. O. Abdelkader in [5] to the duo-ring case by providing characterizations of Duo-rings. In this note, we give an extension of the MP-dimension notion ...