Introduction. Let A be a noetherian ring. When A is commutative (of finite Krull dimension), A is said to be Gorenstein if its injective dimension is finite. If A has finite global dimension, one says that A is regular. If A is arbitrary, these hypotheses are not sufficient to obtain similar results to those of the commutative case. To remedy this problem, M. Auslander has introduced a suppleme...

Let R be a commutative Noetherian ring of prime characteristic p. Assume R (or, more generally, a finitely generated R-module N with SuppR(N) = Spec(R)) has finite F-representation type (abbreviated FFRT) by finitely generated R-modules. Then, for every c ∈ R◦, there is a uniform test exponent Q = p for c and for all R-modules. As a consequence, we show the existence of uniform test exponents o...

Let Λ be a quasi k-Gorenstein ring. For each dth syzygy module M in mod Λ (where 0 ≤ d ≤ k − 1), we obtain an exact sequence 0 → B → M ⊕ P → C → 0 in mod Λ with the properties that it is dual exact, P is projective, C is a (d + 1)st syzygy module, B is a dth syzygy of Ext Λ (D(M),Λ) and the right projective dimension of B ∗ is less than or equal to d − 1. We then give some applications of such ...

Let A be a commutative Noetherian ring, and let R = A[X] be the polynomial ring in an infinite collection X of indeterminates over A. Let SX be the group of permutations of X. The group SX acts on R in a natural way, and this in turn gives R the structure of a left module over the left group ring R[SX ]. We prove that all ideals of R invariant under the action of SX are finitely generated as R[...

In this paper we generalize the Macaulay-Buchberger basis theorem to the case, where the residue class polynomial ring over a Noetherian ring is not necessarily a free module. Recently, this theorem has been extended from polynomial rings over fields to rings, when residue class polynomial ring is free in (Francis & Dukkipati, 2014). As an application of this generalization we develop a theory ...

Let A be a commutative Noetherian ring, and let R = A[x1, x2, . . .] be the polynomial ring in an infinite number of variables xi, indexed by the positive integers. Let S∞ be the symmetric group on an infinite number of letters {1, 2, 3, . . .}. The group S∞ gives a natural action on R, and this in turn gives R the structure of a left module over the (left) group ring RS∞. We prove that ideals ...

Abstract. Let R be a quotient ring of a commutative coherent regular ring by a finitely generated ideal. Hovey gave a bijection between the set of coherent subcategories of the category of finitely presented R-modules and the set of thick subcategories of the derived category of perfect R-complexes. Using this isomorphism, he proved that every coherent subcategory of finitely presented R-module...

1. Introduction. We present two closely related results connecting homological dimension theory and the ideal theory of noetherian rings. The first, Proposition 4.1, asserts that the only ideals of finite homological dimension in a local ring whose associated prime ideals all have grade one are of the form aR:bR. The second, Proposition 4.3, asserts that if R is a noetherian integral domain, th...

Let A be a semprime, right noetherian ring equipped with an automorphism α, and let B := A[[y;α]] denote the corresponding skew power series ring (which is also semiprime and right noetherian). We prove that the Goldie ranks of A and B are equal. We also record applications to induced ideals.