The main purpose of this paper is to prove the following theorem. Let R be a noetherian ring and n a nonnegative integer. Then R[X1, ••• ,Xn] is a going-between ring (=GB) iff R is GB and is (n+l)-piecewise catenarian. PACIFIC JOURNAL OF MATHEMATICS Vol. 86, No. 2, 1980 PIECEWISE CATENARIAN AND GOING BETWEEN RINGS

A commutative ring R is said to satisfy property (P) if every finitely generated proper ideal of R admits a non-zero annihilator. In this paper we give some necessary and sufficient conditions that a ring satisfies property (P). In particular, we characterize coherent rings, noetherian rings and P-coherent rings with property (P).

A number of results are proved concerning the Quillen ^-theory K+(S*G) of the skew group ring S*G, where S is a Noetherian ring and G is a finite group of automorphisms of 5. Applications are given to the computation of AT-groups of group algebras and of equivariant /^-theory for affine varieties.

Normal toric varieties over a field or a discrete valuation ring are classified by rational polyhedral fans. We generalize this classification to normal toric varieties over an arbitrary valuation ring of rank one. The proof is based on a generalization of Sumihiro’s theorem to this non-noetherian setting. These toric varieties play an important role for tropicalizations. 2010 Mathematics Subje...

In the theory of modules over commutative rings there are several possibilities of defining associated prime ideals. The usual definition of an associated prime ideal p for a module M is that p is the annihilator of an element of M . In [2] §1 exercise 17 a generalization of this notion is given. p is called weakly associated (faiblement associé) to M if p is minimal in the set of the prime ide...

Let I be a divisorial ideal of a strongly F-regular ring A. K.-i.Watanabe raised thequestionwhether the symbolicRees algebraRs(I) = ⊕n≥0I is Cohen-Macaulay whenever it is Noetherian. We develop the notion ofmulti-symbolic Rees algebras and use this to show thatRs(I) is indeedCohen-Macaulaywhenever a certain auxiliary ring is finitely generated over A.

Let F = {Fn} be a multiplicative filtration of a local ring such that the Rees algebra R(F) is Noetherian. We recall Burch’s inequality for F and give an upper bound of the a-invariant of the associated graded ring a(G(F)) using a reduction system of F . Applying those results, we study the symbolic Rees algebra of certain ideals of dimension 2.