Steps in Commutative Algebra

Abstract

This introductory account of commutative algebra is aimed at advanced undergraduates and first year graduate students. Assuming only basic abstract algebra, it provides a good foundation in commutative ring theory, from which the reader can proceed to more advanced works in commutative algebra and algebraic geometry. The style throughout is rigorous but concrete, with exercises and examples given within chapters, and hints provided for the more challenging problems used in the subsequent development. After reminders about basic material on commutative rings, ideals and modules are extensively discussed, with applications including to canonical forms for square matrices. The core of the book discusses the fundamental theory of commutative Noetherian rings. Affine algebras over fields, dimension theory and regular local rings are also treated, and for this second edition two further chapters, on regular sequences and Cohen–Macaulay rings, have been added. This book is ideal as a route into commutative algebra.

Generalized star configurations and the Tutte polynomial

“…We denote Min u (I a (C)) to be the set of all distinct minimal prime ideals over I a (C) of height u. The next result presents the primary components of lowest height of I a (C), which are always unique in any minimal primary decomposition of I a (C) (see [13,Theorem 4.29]). Proposition 2.3.…”

Co-Coatomically Supplemented Modules

“…Let R be an By R S we denote the localization of R at the multiplicatively closed subset S of R (see (Sharp, 2000)). …”

Small Prime Modules and Small Prime Submodules

“…Let K = (12) (2) in the Z-module Z 24 . N is a small prime submodule of Z 24 and N = (6) (8)  , while (8) is not a small prime submodule of N, since (0) and (12) are the only small submodules of N with [ (8) :N] = 4Z and 0 2 12, (8)    but 2[ 8 :N] and 12 (8)  . 9.…”