Regular Ideals Research Articles

Regular ideals in commutative rings, sublattices of regular ideals, and Prüfer rings

In this paper we investigate regularly generated, regular, semiregular, and faithful ideals in a commutative ring R and the sublattices they determine. Connections with multiplicative lattice theory are given. Given a Prüfer ring R we show that there is a Prüfer domain D with the sublattice of regular ideals of R isomorphic to the lattice of ideals of D. Numerous examples of rings with zero divisors having certain properties are given. A Prüfer ring with an invertible ideal that is not generated by regular elements is constructed. An example is given to show that the intersection of two regular principal ideals need not be generated by regular elements.

Relatively regular ideals in semiprime Banach algebras

Nella prima parte di questo lavoro vengono studiate le relazioni che intercorrono tra il concetto di relativa regolarita in un'algebra non necessariamente unitaria ed alcuni ideali (radicale, zoccolo, prezoccolo) che intervengono nello sviluppo della teoria di Fredholm in algebre di Banach. Nella seconda parte del lavoro viene dimostrato che in certe algebre di BanachA complesse semiprime l'esistenza di un ideale relativamente regolare chiuso implica cheA e unitaria e di dimensione finita.

Characterizing prüfer rings via their regular ideals

Characterizing prüfer rings via their regular ideals

High order derivations and primary ideals to regular prime ideals

High order derivations and primary ideals to regular prime ideals

Completely semiprime and Abelian regular ideals of a ring

Completely semiprime and Abelian regular ideals of a ring

Regular ideals and commuting positive operators

A partially ordered vector space over the reals R is a vector space V with a cone C which is proper in the sense that Cn -C = (0) and with the partial order given by saying x 0 such that -/?e < x < /?e. V is said to be positively generated if V= C C. An order ideal is a linear subspace J such that x < y ,< z with x, z E J implies 4’ E J. The ideal J is called regular if there exists an element a E V such that the coset [u] is an order unit for V/J. If V is also a normed vector space we say C is normal if there exists a positive number y such that ]]x +v]] < y ]]x]] for all X,J’ E C. A linear operator T on V is said to be positive if x E C implies T(x) E C. The following result is proved in [ l].

Invariant regular ideals in Brown-Peterson homology

Invariant regular ideals in Brown-Peterson homology

A Shorter Proof of Goldie's Theorem

In this note we present an extremely short proof of Goldie's theorem on the structure of semiprime Noetherian rings [1]. The outline of the proof was given by Procesi and Small in [4]. By utilizing the concept of the singular ideal of a ring we have been able to weaken the hypotheses of many of the steps in [4]. Most significantly, we are able to avoid a reduction to the case of prime rings, and in Lemma 5 we give an informative list of the relationship between regular elements and essential ideals of semiprime rings.

On the Annihilators of the Injective Hull of a Module

In [2, page 151], J. Lambek proposes the following exercise: With any maximal right ideal M of a ring R with 1 associate the ideal . Show that OM, is the right annihilator of the injective hull of the right R-module R/M. The purpose of this note is to show that the above statement is true for a much larger class of right ideals than that of maximal regular right ideals of a ring.

On Maximal Regular Ideals and Identities in the Tensor Product of Commutative Banach Algebras

Let A1and A2be commutative Banach algebras and A1 ⊙ A2 their algebraic tensor product over the complex numbers C.There is always a t least one norm, namely the greatest cross-norm γ (2), on A1 ⊙ A2 that renders it a normed algebra. We shall write A1 ⊗αA2 for the α-completion of A1⊙ A2when αis an algebra norm on A1⊙ A2.Gelbaum (2; 3), Tomiyama (9), and Gil de Lamadrid (4) have shown that for certain algebra norms α on A1⊙ A2 every complex homomorphism on A1 ⊙ A2 is α-continuous. In § 3 of this paper, we present a condition on an algebra norm α which is equivalent to the α-continuity of every complex homomorphism on A1⊙ A2.

Containment relations between classes of regular ideals in a ring with few zero divisors

Containment relations between classes of regular ideals in a ring with few zero divisors

Regular Banach Algebras with a Countable Space of Maximal Regular Ideals

Regular Banach Algebras with a Countable Space of Maximal Regular Ideals

Regular Ideals of Partially Ordered Vector Spaces

Proceedings of the London Mathematical SocietyVolume s3-6, Issue 4 p. 626-640 Articles Regular Ideals of Partially Ordered Vector Spaces F. F. Bonsall, F. F. Bonsall Durham University, Newcastle-on-TyneSearch for more papers by this author F. F. Bonsall, F. F. Bonsall Durham University, Newcastle-on-TyneSearch for more papers by this author First published: October 1956 https://doi.org/10.1112/plms/s3-6.4.626Citations: 5AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Citing Literature Volumes3-6, Issue4October 1956Pages 626-640 RelatedInformation

Regular Banach algebras with a countable space of maximal regular ideals

Regular Banach algebras with a countable space of maximal regular ideals

The group algebra of a locally compact group

Introduction. This paper concerns the theory of ideals in the algebra (called the group algebra) of all complex-valued functions on a locally compact (abbreviated to LC) group which are integrable with respect to Haar measure, multiplication being defined as convolution. It is proved that the group algebra of a group which is either LC abelian or compact is semisimple, an algebra being called semi-simple in case the intersection of all regular maximal ideals is the null ideal (a regular ideal is defined as one modulo which the algebra has an identity). This extends the theorem that the group algebra of a finite group is semi-simple. A weaker kind of semi-simplicity is proved in the case of a general LC group, an algebra being called weakly semi-simple when both the intersection of all regular maximal right ideals is the null ideal and the same for left ideals. (In the case of a finite-dimensional algebra these concepts of semi-simplicity are equivalent to that of Wedderburn, and in the case of a commutative Banach algebra with an identity-which Gelfand designates as a normed ring-are equivalent to Gelfand's concept of the vanishing of the radical of the algebra.) It is shown that an ideal in a (strongly) semi-simple algebra can be resolved into regular ideals in an approximate fashion, the approximation being in terms of a topology which is (algebraically) introduced into the familv of all regular maximal ideals. This topologized family (which is called the spectrum of the algebra, this term being suggested by the case of the algebra generated by a linear operator) is determined in explicit fashion for the group algebra of a group which is either LC abelian or compact, and partially determined when the group is discrete. Specifically, in the case of a group which is respectively either LC abelian, compact, or discrete, the spectrum of the group algebra is homeomorphic with the group of continuous characters, discrete or compact. A closed ideal in a group algebra is the kernel of a kind of extension from the group to the group algebra of a bounded strongly continuous representation of the group by linear operators on a Banach space. If the