for making many comments and corrections concerning these notes. All rings are commutative and contain multiplicative identity, moreover we will always insist that ring homomorphisms respect the multiplicative identity element. Local rings are assumed to be Noetherian. Additionally, all modules are unitary modules. We have made an attempt to be consistent with our notation: (1) Rings are often ...

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).

In this paper, we extend the well-known Hilbert’s syzygy theorem to the Gorenstein homological dimensions of rings. Also, we study the Gorenstein homological dimensions of direct products of rings. Our results generate examples of non-Noetherian rings of finite Gorenstein dimensions and infinite classical weak dimension.

CARL FAITH All rings considered are commutative with unit. A ring R is SISI (in Vámos' terminology) if every subdirectly irreducible factor ring R/I is self-injective . SISI rings include Noetherian rings, Morita rings, and almost maximal valuation rings ([Vil) . In [F3] we raised the question of whether a polynomial ring R[-1 over a SISI ring R is again SISI . In this paper we show this is not...

Articles on the history of mathematics can be written from many different perspectives. Some aim to survey a more or less wide landscape, and require the observer to watch from afar as theories develop and movements are born or become obsolete. At the other extreme, there are those that try to shed light on the history of particular theorems and on the people who created them. This article belo...

We generalize two major ways of obtaining derived equivalences, the tilting process by Happel, Reiten and Smalø and Happel’s Tilting Theorem, to the setting of finitely presented modules over right coherent rings. Moreover, we extend the characterization of quasi–tilted artin algebras as the almost hereditary ones to all right noetherian rings. We also give a streamlined and general presentatio...

we introduce the class of “right almost v-rings” which is properly between the classes of right v-rings and right good rings. a ring r is called a right almost v-ring if every simple r-module is almost injective. it is proved that r is a right almost v-ring if and only if for every r-module m, any complement of every simple submodule of m is a direct summand. moreover, r is a right almost v-rin...

The Generalized Principal Ideal Theorem is one of the cornerstones of dimension theory for Noetherian rings. For an R-module M, we identify certain submodules of M that play a role analogous to that of prime ideals in the ring R. Using this definition, we extend the Generalized Principal Ideal Theorem to modules.

We prove over some local commutative noetherian rings that the sequence of Betti numbers of every finitely generated module is either eventually constant or has termwise exponential growth.