Some Commutative Algebra

2017 ◽  
Author(s):  
Matthias Aschenbrenner ◽  
Lou van den Dries ◽  
Joris van der Hoeven
Keyword(s):  
Commutative Algebra ◽  
Principal Ideal ◽  
Noetherian Rings ◽  
Local Rings ◽  
Zariski Topology ◽  
Algebraic Differential Equations ◽  
Field Extensions ◽  
Notational Convention ◽  
Kähler Differentials ◽  
Integrally Closed

This chapter provides a background on commutative algebra and gives a self-contained proof of Johnson's Theorem 5.9.1 on regular solutions of systems of algebraic differential equations. It presents the facts on regular local rings and Kähler differentials needed for Theorem 5.9.1. It also recalls a common notational convention concerning a commutative ring R and an R-module M, with U and V as additive subgroups of R and M. Other topics include the Zariski topology, noetherian rings and spaces, rings and modules of finite length, integral extensions and integrally closed domains, Krull's Principal Ideal Theorem, differentials, and derivations on field extensions.

Some results concerning the local analytic branches of an algebraic variety

1953 ◽  
Vol 49 (3) ◽  
pp. 386-396 ◽  
Author(s):  
D. G. Northcott
Keyword(s):  
Algebraic Geometry ◽  
Power Series ◽  
Algebraic Variety ◽  
Recent Progress ◽  
Rapid Development ◽  
Noetherian Rings ◽  
Local Rings ◽  
Power Series Rings ◽  
Analytic Algebra ◽  
Topological Theories

The recent progress of modern algebra in analysing, from the algebraic standpoint, the foundations of algebraic geometry, has been marked by the rapid development of what may be called ‘analytic algebra’. By this we mean the topological theories of Noetherian rings that arise when one uses ideals to define neighbourhoods; this includes, for instance, the theory of power-series rings and of local rings. In the present paper some applications are made of this kind of algebra to some problems connected with the notion of a branch of a variety at a point.

scholarly journals NORMAL DOMAINS WITH MONOMIAL PRESENTATIONS

2009 ◽  
Vol 19 (03) ◽  
pp. 287-303 ◽  
Author(s):  
ISABEL GOFFA ◽  
ERIC JESPERS ◽  
JAN OKNIŃSKI
Keyword(s):  
Commutative Algebra ◽  
Class Group ◽  
Semigroup Algebra ◽  
Closed Domain ◽  
Finitely Generated ◽  
Defining Relations ◽  
Integrally Closed ◽  
Algebra Over A Field

Let A be a finitely generated commutative algebra over a field K with a presentation A = K 〈X1,…, Xn | R〉, where R is a set of monomial relations in the generators X1,…, Xn. So A = K[S], the semigroup algebra of the monoid S = 〈X1,…, Xn | R〉. We characterize, purely in terms of the defining relations, when A is an integrally closed domain, provided R contains at most two relations. Also the class group of such algebras A is calculated.

scholarly journals PRÜFER ⋆-MULTIPLICATION DOMAINS AND SEMISTAR OPERATIONS

2003 ◽  
Vol 02 (01) ◽  
pp. 21-50 ◽  
Author(s):  
M. FONTANA ◽  
P. JARA ◽  
E. SANTOS
Keyword(s):  
Characterization Theorem ◽  
Field Extensions ◽  
Star Operation ◽  
Classical Characterization ◽  
Integrally Closed ◽  
Prufer Domains ◽  
Semistar Operation ◽  
Prüfer Domains ◽  
The Given

Starting from the notion of semistar operation, introduced in 1994 by Okabe and Matsuda [49], which generalizes the classical concept of star operation (cf. Gilmer's book [27]) and, hence, the related classical theory of ideal systems based on the works by W. Krull, E. Noether, H. Prüfer, P. Lorenzen and P. Jaffard (cf. Halter–Koch's book [32]), in this paper we outline a general approach to the theory of Prüfer ⋆-multiplication domains (or P⋆MDs), where ⋆ is a semistar operation. This approach leads to relax the classical restriction on the base domain, which is not necessarily integrally closed in the semistar case, and to determine a semistar invariant character for this important class of multiplicative domains (cf. also J. M. García, P. Jara and E. Santos [25]). We give a characterization theorem of these domains in terms of Kronecker function rings and Nagata rings associated naturally to the given semistar operation, generalizing previous results by J. Arnold and J. Brewer ]10] and B. G. Kang [39]. We prove a characterization of a P⋆MD, when ⋆ is a semistar operation, in terms of polynomials (by using the classical characterization of Prüfer domains, in terms of polynomials given by R. Gilmer and J. Hoffman [28], as a model), extending a result proved in the star case by E. Houston, S. J. Malik and J. Mott [36]. We also deal with the preservation of the P⋆MD property by ascent and descent in case of field extensions. In this context, we generalize to the P⋆MD case some classical results concerning Prüfer domains and PvMDs. In particular, we reobtain as a particular case a result due to H. Prüfer [51] and W. Krull [41] (cf. also F. Lucius [43] and F. Halter-Koch [34]). Finally, we develop several examples and applications when ⋆ is a (semi)star given explicitly (e.g. we consider the case of the standardv-, t-, b-, w-operations or the case of semistar operations associated to appropriate families of overrings).

Hilbert rings and G(oldman)-rings issued from amalgamated algebras

2018 ◽  
Vol 17 (02) ◽  
pp. 1850023 ◽  
Author(s):  
L. Izelgue ◽  
O. Ouzzaouit
Keyword(s):  
New York ◽  
Commutative Algebra ◽  
Quotient Ring ◽  
Sufficient Conditions ◽  
Ring Homomorphism ◽  
Necessary And Sufficient Conditions ◽  
Local Rings ◽  
Finitely Generated ◽  
Necessary And Sufficient ◽  
The Way

Let [Formula: see text] and [Formula: see text] be two rings, [Formula: see text] an ideal of [Formula: see text] and [Formula: see text] be a ring homomorphism. The ring [Formula: see text] is called the amalgamation of [Formula: see text] with [Formula: see text] along [Formula: see text] with respect to [Formula: see text]. It was proposed by D’anna and Fontana [Amalgamated algebras along an ideal, Commutative Algebra and Applications (W. de Gruyter Publisher, Berlin, 2009), pp. 155–172], as an extension for the Nagata’s idealization, which was originally introduced in [Nagata, Local Rings (Interscience, New York, 1962)]. In this paper, we establish necessary and sufficient conditions under which [Formula: see text], and some related constructions, is either a Hilbert ring, a [Formula: see text]-domain or a [Formula: see text]-ring in the sense of Adams [Rings with a finitely generated total quotient ring, Canad. Math. Bull. 17(1) (1974)]. By the way, we investigate the transfer of the [Formula: see text]-property among pairs of domains sharing an ideal. Our results provide original illustrating examples.

CHARACTERIZATIONS OF WEAK LIFTING AND LIFTING MODULES

2007 ◽  
Vol 06 (02) ◽  
pp. 267-279 ◽  
Author(s):  
RACHID TRIBAK
Keyword(s):  
Noetherian Rings ◽  
Local Rings

The aim of this paper is to prove that the study of lifting and weak lifting modules over commutative noetherian rings reduces to the case of coatomic modules, radical artinian Kirby-p-coprimary modules and radical minimax Kirby-p-coprimary modules with [Formula: see text] over local rings.

scholarly journals On the Ideal Case of a Conjecture of Huneke and Wiegand

2019 ◽  
Vol 62 (3) ◽  
pp. 847-859 ◽  
Author(s):  
Olgur Celikbas ◽  
Shiro Goto ◽  
Ryo Takahashi ◽  
Naoki Taniguchi
Keyword(s):  
Free Module ◽  
Affirmative Answer ◽  
Noetherian Rings ◽  
Ideal Case ◽  
One Dimensional ◽  
Higher Dimensional ◽  
Integrally Closed ◽  
Integrally Closed Ideals ◽  
Torsion Free ◽  
The Ideal

AbstractA conjecture of Huneke and Wiegand claims that, over one-dimensional commutative Noetherian local domains, the tensor product of a finitely generated, non-free, torsion-free module with its algebraic dual always has torsion. Building on a beautiful result of Corso, Huneke, Katz and Vasconcelos, we prove that the conjecture is affirmative for a large class of ideals over arbitrary one-dimensional local domains. Furthermore, we study a higher-dimensional analogue of the conjecture for integrally closed ideals over Noetherian rings that are not necessarily local. We also consider a related question on the conjecture and give an affirmative answer for first syzygies of maximal Cohen–Macaulay modules.

Fibre products of Noetherian rings and their applications

1985 ◽  
Vol 97 (2) ◽  
pp. 231-241 ◽  
Author(s):  
Tetsushi Ogoma
Keyword(s):  
Commutative Algebra ◽  
Ring Theory ◽  
Noetherian Rings ◽  
Fibre Products ◽  
Fibre Product

The notion of fibre product in a category is quite basic and has been studied by many authors. Also in ring theory, it is known that the fibre product is useful in the construction of examples. (See for example [3], [4] and references of [1].) Unfortunately, most such examples are non-noetherian and so are unsatisfactory from the viewpoint of commutative algebra.

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