REGULARITY AND STRONG REGULARITY IN THE CONTEXT OF CERTAIN CLASSES OF RINGS

We consider the ring R[x]/(xn+1), where R is a ring, R[x] is the ring of polynomials in an indeterminant x, (xn+1) is the ideal of R[x] generated by xn+1 and n is a positive integer. The aim of this paper is to show that regularity or strong regularity of a ring R is necessary and sufficient condition under which the ring R[x]/(xn+1) is an example of a ring which belongs to some important classes of rings. In this context, we discuss distributive rings, Bézout rings, Gaussian rings, quasi-morphic rings, semihereditary rings, and rings which have weak dimension less than or equal to one.

REGULAR LOCAL ALGEBRAS OVER VALUATION DOMAINS: WEAK DIMENSION AND REGULAR SEQUENCES

A local ring O is called regular if every finitely generated ideal I ◃ O possesses finite projective dimension. In the article localizations O = Aq, q ∈ Spec A, of a finitely presented, flat algebra A over a Prüfer domain R are investigated with respect to regularity: this property of O is shown to be equivalent to the finiteness of the weak homological dimension wdim O. A formula to compute wdim O is provided. Furthermore regular sequences within the maximal ideal M ◃ O are studied: it is shown that regularity of O implies the existence of a maximal regular sequence of length wdim O. If q ∩ R has finite height, then this sequence can be chosen such that the radical of the ideal generated by its members equals M. As a consequence it is proved that if O is regular, then the factor ring O/(q ∩ R)O, which is noetherian, is Cohen–Macaulay. If in addition (q ∩ R)Rq ∩ R is not finitely generated, then O/(q ∩ R)O itself is regular.

An equivalence induced by Ext and Tor applied to the finitistic weak dimension of coherent rings

Let R be a ring, see below for other notation. The functor categories (mod-R, Ab) and ((R-mod)op, Ab) have received considerable attention since the 1960s. The first of these has achieved prominence in the model theory of modules and most particularly in the investigation of the representation theory of Artinian algebras. Both [11, Chapter 12] and [8] contain accounts of the use (mod-R, Ab) may be put to in the model theoretic setting, and Auslander's review, [1], details the application of (mod-R, Ab) to the study of Artinian algebras. The category ((R-mod)op, Ab) has been less fully exploited. Much work, however, has been devoted to the study of the transpose functor between R-mod and mod-R. Warfield's paper, [13], describes this for semiperfect rings, and this duality is an essential component in the construction of almost split sequences over Artinian algebras, see [4]. In comparison, the general case has been neglected. This paper seeks to remedy this situation, giving a concrete description of the resulting equivalence between (mod-R, Ab) and ((R-mod)op, Ab) for an arbitrary ring R.