The small finitistic dimensions over commutative rings

Let $R$ be a commutative ring with identity. The small finitistic dimension \fPD(R) of $R$ is defined to be the supremum of projective dimensions of $R$-modules with finite projective resolutions. In this paper, we characterize a ring $R$ with \fPD(R)\leq n using finitely generated semi-regular ideals, tilting modules, cotilting modules of cofinite type or vaguely associated prime ideals. As an application, we obtain that if $R$ is a Noetherian ring, then \fPD(R)= \sup\{\grade(\m,R)|\m\in \Max(R)\} where \grade(\m,R) is the grade of \m on $R$ . We also show that a ring $R$ satisfies \fPD(R)\leq 1 if and only if $R$ is a \DW ring. As applications, we show that the small finitistic dimensions of strong \Prufer\ rings and \LPVDs are at most one. Moreover, for any given $n\in \mathbb{N}$, we obtain examples of total rings of quotients $R$ with \fPD(R)=n

On Strong Mori domains

AbstractIn this note we investigate properties of Strong Mori domains, which form a proper subclass of Mori domains. In particular, we show that Strong Mori domains satisfy the Principal Ideal Theorem, the Hubert Basis Theorem and the Krull Intersection Theorem. We also provide some new characterizations of Krull domains and show that the complete integral closure of a Strong Mori domain is a Krull domain

Foundations of commutative rings and their modules

This book provides an introduction to the basics and recent developments of commutative algebra. A glance at the contents of the first five chapters shows that the topics covered are ones that usually are included in any commutative algebra text. However, the contents of this book differ significantly from most commutative algebra texts: namely, its treatment of the Dedekind–Mertens formula, the (small) finitistic dimension of a ring, Gorenstein rings, valuation overrings and the valuative dimension, and Nagata rings. Going further, Chapter 6 presents w-modules over commutative rings as they can be most commonly used by torsion theory and multiplicative ideal theory. Chapter 7 deals with multiplicative ideal theory over integral domains. Chapter 8 collects various results of the pullbacks, especially Milnor squares and D+M constructions, which are probably the most important example-generating machines. In Chapter 9, coherent rings with finite weak global dimensions are probed, and the local ring of weak global dimension two is elaborated on by combining homological tricks and methods of star operation theory. Chapter 10 is devoted to the Grothendieck group of a commutative ring. In particular, the Bass–Quillen problem is discussed. Finally, Chapter 11 aims to introduce relative homological algebra, especially where the related concepts of integral domains which appear in classical ideal theory are defined and investigated by using the class of Gorenstein projective modules. Each section of the book is followed by a selection of exercises of varying degrees of difficulty. This book will appeal to a wide readership from graduate students to academic researchers who are interested in studying commutative algebra

Indecomposable, projective and flat S-posets

Abstract. For a monoid S, a (left) S-act is a non-empty set B together with a mapping S × B → B sending (s, b) to sb such that s(tb) = (st)b and 1b = b for all s, t ∈ S and b ∈ B. Right S-acts A can also be defined, and a tensor product A ⊗S B (a set) can be defined that has the customary universal property with respect to balanced maps from A×B into arbitrary sets. Over the past three decades, an extensive theory of flatness properties has been developed (involving free and projective acts, and flat acts of various sorts, defined in terms of when the tensor product functor has certain preservation properties). A recent and complete discussion of this area is contained in the monograph Monoids, Acts and Categories by M. Kilp et al. (Walter de Gruyter, New York, 2000). To date, there have been only a few attempts to generalize this material to ordered monoids acting on partially ordered sets (S-posets). The present paper is devoted to such a generalization. A unique decomposition theorem for S-posets is given, based on strongly convex, indecomposable S-subposets, and a structure theorem for projective S-posets is given. A criterion for when two elements of the tensor product of S-posets is given, which is then applied to investigate several flatness properties. 1. introduction and preliminaries If S is a monoid, it is well-known that S-acts (also called S-sets, S-systems, S-automata, etc.) play an important role not only in studying properties of monoids, but also in other mathematical areas, such as graph theory and algebraic automata theory (see [1], [2]). Flatness and projectivity are important topics in the study of acts over monoids. Flat acts are important in studying amalgamation properties of monoids (see [1] and references contained therein), and projective and indecomposable acts play an essential role in studying th