Homologically Homogeneous Rings

Abstract

Abstract. In this paper we study the structure of a right Noetherian ring R of finite right global dimesion integral over a central subring C and satisfying the following condition:if V, W are irreducible right K-modules with rc(V) = rc(W) then prdim(K) = pràim(W).1. Introduction. Let £ be a ring with a central subring C. We shall say that R is homologically homogeneous (horn, horn.) over C if (i) R is right Noetherian; (ii) R is integral over C; (iii) the right global (projective) dimension of R (denoted rtgldim(£)) is finite; and (iv) if V and M are irreducible right £-modules whose annihilators in C, namely rc(V) and rc(W), are equal then prdimR(K) = prdim^H7). . In addition, we show in §7 that polynomial rings over horn. horn, rings, and certain group rings, are horn. horn.Rings of type (c) were studied in [5] where we showed that their maximal ideals had rank equal to their grade, defined in terms of maximal lengths of C-sequences. We termed such rings C-Macaulay-a precise definition is given in §2-and showed in [5, §4] that they share many of the properties of commutative Cohen-Macaulay

Smooth polynomial identity algebras with almost factorial centers

“…Let X = X(P ) be such a typical clique, with P a height one prime in R. It suffices to show that X = {P }, namely that P is localizable. Now by [5,Theorem 3.5] R X is hereditary and consequently P X is a projective R X -module (from both sides). By assumption and Corollary 8 (working with R S , where S ≡ C(X)) we get that P X is stably free.…”

“…Now by [5,Theorem 4.1] R/dR has an Artinian quotient ring,P = P /dR is the unique minimal prime ideal and if s / ∈ q is central, thens is regular inR = R/dR. Now R q is Azumaya, so P q = q q R q = dR q .…”

“…is a smooth Noetherian PI ring and hence by[5, Theorem 4.1(ii)] R[t]/hR[t]has an Artinian quotient ring.Now (t 1 ), . .…”

On symmetric, smooth and Calabi–Yau algebras

“…The fact that R is smooth implies by [8], that R is a CM Z(R)-module and consequently R is a finitely generated free C-module. This was the only required property of C in the proof of Corollaries D and G. 2 .…”

“…Recall that R is called smooth if gl.dim R is finite and the projective dimension of all simple R-modules is constant along each clique (e.g. see [4,8] and [27]). We shall then show that many of the smooth PI algebras appearing in "nature" are in fact CY algebras.…”