Showing papers on "Global dimension published in 1977"

Noetherian fixed rings.

Abstract: by a group of automorphisms of R. This paper explores what happens when the group is finite and the fixed ring S is assumed to be Noetherian Easy examples show that R may not be Noetherian; however, in this paper it is shown that R is Noetherian with some rather natural assuptions. More precisely we prove the Theorem 2: Let S be a semiprime ring. Assume that G is a finite group of automorphisms of 5 and that S has no | G [-torsion. If S° is left noetherian then S is left noetherian.

On almost complete intersections

Abstract: Let R be a local ring such that R=S/I where S is a regular local ring and I is a prime ideal of height r. In this paper it is shown that if I is minimally generated by r+1 elements, then there exists an R-homomorphism φ: KR→Rr+1 such that φ is an injection and Rr+1/φ(KR)≌I/I2 where KR:=ExtSr(R,S) the canonical module of R. Moreover, in case where S is a locality over a perfect field k, it is also shown that if R is Cohen-Macaulay and I2 is a primary ideal, then the homological dimension of the differential module ΩR/k is infinite.

Krull dimension in graded rings

Abstract: (1977). Krull dimension in graded rings. Communications in Algebra: Vol. 5, No. 3, pp. 319-329.

The homological dimension of a torsion-free abelian group of finite rank as a module over its ring of endomorphisms

Abstract: L’accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova » (http://rendiconti.math.unipd.it/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

On normality of a noetherian ring

Abstract: In the present article, we understand by a ring a commutative ring with identity; any local or semi-local rings are assumed to be noetherian. A ring is called q u a s i-n o rm a l' if it is integrally closed in its total quotient ring. If such a ring has no nilpotent elements (except zero), then it is the direct sum of some normal domains under certain finiteness condition, and several results are known in such a case. Our main result is a characterization of a quasi-normal noetherian ring, which asserts as follows :2 ) A noetherian ring R is quasi-normal if and only if the following two conditions are satisfied : (1) If P is a prime ideal of height one in R , and if P contains a non-zerodivisor, then R , is a discrete valuation ring. (2) If a non-unit a of R is not a zero-divisor, then a R has no imbedded prime divisor. As for normal rings (under any one of the definitions in foot-note 1), the normality is carried over to its rings of quotients. But quasi-normality may not be carried over to rings of quotients. The present article deals also with some topics related to this fact. For a ring R , Q (R ) denotes the total quotient ring of R.