scholarly journals On the radical of a category

1964 ◽  
Vol 4 (3) ◽  
pp. 299-307 ◽  
Author(s):  
G. M. Kelly
Keyword(s):  
Jacobson Radical ◽  
Additive Category ◽  
The Ideal

In [1] the concept of completeness of a functor was introduced and, in the cse of additive * categories and and an additive functor T: → , a criterion for T (supposed surjective) to be complete was given in terms of the kernel of T: this was that for each object A of the ideal A should be containded in the (Jacobson) radical of A. (The meaning of this notation and nomemclature is recalled in § 2 below). The question arises whether in any additive category there is a greatest ideal with this property, so that the canonical functor T: → / is in some sense the coarsest that faithfully represents the objects (but not the maps) of .

Noetherian Tensor Products

1972 ◽  
Vol 15 (2) ◽  
pp. 235-238
Author(s):  
E. A. Magarian ◽  
J. L. Motto
Keyword(s):  
Tensor Product ◽  
Jacobson Radical ◽  
Tensor Products ◽  
Minimum Condition ◽  
Ideal Structure ◽  
The Ideal

Relatively little is known about the ideal structure of A⊗RA' when A and A' are R-algebras. In [4, p. 460], Curtis and Reiner gave conditions that imply certain tensor products are semi-simple with minimum condition. Herstein considered when the tensor product has zero Jacobson radical in [6, p. 43]. Jacobson [7, p. 114] studied tensor products with no two-sided ideals, and Rosenberg and Zelinsky investigated semi-primary tensor products in [9].All rings considered in this paper are assumed to be commutative with identity. Furthermore, R will always denote a field.

Radical Regularity in Differential Rings

1971 ◽  
Vol 23 (2) ◽  
pp. 197-201 ◽  
Author(s):  
Howard E. Gorman
Keyword(s):  
Jacobson Radical ◽  
Regular Ring ◽  
Prime Ideals ◽  
Differential Ring ◽  
Quotient Maps ◽  
The Stability ◽  
Definition Of ◽  
Finitely Generated Modules ◽  
Regular Rings ◽  
The Ideal

In [1], we discussed completions of differentially finitely generated modules over a differential ring R. It was necessary that the topology of the module be induced by a differential ideal of R and it was natural that this ideal be contained in J(R), the Jacobson radical of R. The ideal to be chosen, called Jd(R), was the intersection of those ideals which are maximal among the differential ideals of R. The question as to when Jd(R) ⊆ J(R) led to the definition of a class of rings called radically regular rings. These rings do satisfy the inclusion, and we showed in [1, Theorem 2] that R could always be “extended”, via localization, to a radically regular ring in such a way as to preserve all its differential prime ideals.In the present paper, we discuss the stability of radical regularity under quotient maps, localization, adjunction of a differential indeterminate, and integral extensions.

Annihilator-small Right Ideals

2011 ◽  
Vol 18 (spec01) ◽  
pp. 785-800
Author(s):  
W. K. Nicholson ◽  
Yiqiang Zhou
Keyword(s):  
Jacobson Radical ◽  
Singular Ideal ◽  
Left Annihilator ◽  
The Ideal

A right ideal A of a ring R is called annihilator-small if A+T=R, T a right ideal, implies that [Formula: see text], where [Formula: see text] indicates the left annihilator. The sum Ar of all such right ideals turns out to be a two-sided ideal that contains the Jacobson radical and the left singular ideal, and is contained in the ideal generated by the total of the ring. The ideal Ar is studied, conditions when it is annihilator-small are given, its relationship to the total of the ring is examined, and its connection with related rings is investigated.

A Generalization of Semiregular and Almost Principally Injective Rings

2010 ◽  
Vol 17 (spec01) ◽  
pp. 905-916
Author(s):  
A. Çiğdem Özcan ◽  
Pınar Aydoğdu
Keyword(s):  
Jacobson Radical ◽  
Left And Right ◽  
Singular Ideal ◽  
The Right ◽  
The Ideal

In this article, we call a ring R right almost I-semiregular for an ideal I of R if for any a ∈ R, there exists a left R-module decomposition lRrR(a) = P ⊕ Q such that P ⊆ Ra and Q ∩ Ra ⊆ I, where l and r are the left and right annihilators, respectively. This generalizes the right almost principally injective rings defined by Page and Zhou, I-semiregular rings defined by Nicholson and Yousif, and right generalized semiregular rings defined by Xiao and Tong. We prove that R is I-semiregular if and only if for any a ∈ R, there exists a decomposition lRrR(a) = P ⊕ Q, where P = Re ⊆ Ra for some e2 = e ∈ R and Q ∩ Ra ⊆ I. Among the results for right almost I-semiregular rings, we show that if I is the left socle Soc (RR) or the right singular ideal Z(RR) or the ideal Z(RR) ∩ δ(RR), where δ(RR) is the intersection of essential maximal left ideals of R, then R being right almost I-semiregular implies that R is right almost J-semiregular for the Jacobson radical J of R. We show that δl(eRe) = e δ(RR)e for any idempotent e of R satisfying ReR = R and, for such an idempotent, R being right almost δ(RR)-semiregular implies that eRe is right almost δl(eRe)-semiregular.

scholarly journals Modules over polycyclic groups have many irreducible images

1981 ◽  
Vol 22 (2) ◽  
pp. 141-150 ◽  
Author(s):  
Kenneth A. Brown
Keyword(s):  
Finite Group ◽  
Group Ring ◽  
Noetherian Ring ◽  
Jacobson Radical ◽  
Finitely Generated ◽  
Left Module ◽  
Image Position ◽  
Polycyclic Groups ◽  
Left Annihilator ◽  
The Ideal

Recall that a Noetherian ring R is a Hilbert ring if the Jacobson radical of every factor ring of R is nilpotent. As one of the main results of [13], J. E. Roseblade proved that if J is a commutative Hilbert ring and G is a polycyclic-by-finite group then JG is a Hilbert ring. The main theorem of this paper is a generalisation of this result in the case where all the field images of J are absolute fields—we shall say that J is absolutely Hilbert. The result is stated in terms of the (Gabriel–Rentschler–) Krull dimension; the definition and basic properties of this may be found in [5]. Let M be a finitely generated right module over the ring R. We write AnnR(M) (or just Ann(M)) for the ideal {r ∈ R: Mr = 0}, the annihilator of M in R. If M is also a left module, its left annihilator will be denoted l-AnnR(M). If R is a group ring JG, put

scholarly journals On the divisor class groups of a two-dimensional local ring and its form ring

1988 ◽  
Vol 110 ◽  
pp. 137-149 ◽  
Author(s):  
Dario Portelli ◽  
Walter Spangher
Keyword(s):  
Local Ring ◽  
Noetherian Ring ◽  
Jacobson Radical ◽  
Integral Domain ◽  
Two Dimensional ◽  
Class Groups ◽  
Divisor Class ◽  
Form Ring ◽  
The Ideal

Let A be a noetherian ring and let I be an ideal of A contained in the Jacobson radical of A: Rad (A). We assume that the form ring of A with respect to the ideal I: G = Gr (A, I), is a normal integral domain. Hence A is a normal integral domain and one can ask for the links between Cl(A) and Cl(G).

scholarly journals Core and dual core inverses of a sum of morphisms

Filomat ◽  
2019 ◽  
Vol 33 (10) ◽  
pp. 2931-2941 ◽  
Author(s):  
Tingting Li ◽  
Jianlong Chen ◽  
Dingguo Wang ◽  
Sanzhang Xu
Keyword(s):  
Jacobson Radical ◽  
The Core ◽  
Unital Ring ◽  
Additive Category ◽  
Core Inverse ◽  
Dual Core

Let C be an additive category with an involution *. Suppose that ? : X ? X is a morphism of C with core inverse ?# : X ? X and ? : X ? X is a morphism of C such that 1X + ?#? is invertible. Let ? = (1X+?#?)-1, ? = (1X+??#)-1, ? = (1X-??#)??(1X-?#?), ? = ?(1X-?#?)?-1??#?,? = ??#??-1(1X-??#)?,? = ?*(?#(*?*(1X-??#)?. Then f = ? + ? ? ? has a core inverse if and only if 1X-?, 1X-? and 1X-? are invertible. Moreover, the expression of the core inverse of f is presented. Let R be a unital *-ring and J(R) its Jacobson radical, if a ? R# with core inverse a # and j ? J(R), then a + j ? R# if and only if (1-aa#)j(1+a#j)-1(1-a#a) = 0. We also give the similar results for the dual core inverse.

Localization of Intracellular Antigens in thin Sections with Ferritin Labeled Antibody

1970 ◽  
Vol 28 ◽  
pp. 174-175
Author(s):  
M.S. Shahrabadi ◽  
T. Yamamoto
Keyword(s):  
Bovine Serum Albumin ◽  
Bovine Serum ◽  
Antigenic Determinants ◽  
Conjugate Prior ◽  
Thin Sections ◽  
Specific Attachment ◽  
Intracellular Antigen ◽  
Antigen Antibody ◽  
Ideal Method ◽  
The Ideal

The technique of labeling of macromolecules with ferritin conjugated antibody has been successfully used for extracellular antigen by means of staining the specimen with conjugate prior to fixation and embedding. However, the ideal method to determine the location of intracellular antigen would be to do the antigen-antibody reaction in thin sections. This technique contains inherent problems such as the destruction of antigenic determinants during fixation or embedding and the non-specific attachment of conjugate to the embedding media. Certain embedding media such as polyampholytes (2) or cross-linked bovine serum albumin (3) have been introduced to overcome some of these problems.

Criteria and Methods for Reliable Three-Dimensional Image Reconstruction

1974 ◽  
Vol 32 ◽  
pp. 330-331
Author(s):  
R. A. Crowther
Keyword(s):  
Image Reconstruction ◽  
Finite Number ◽  
Density Distribution ◽  
Electron Micrographs ◽  
Mathematical Problem ◽  
Three Dimensional ◽  
Ideal Case ◽  
Dimensional Image ◽  
General Object ◽  
The Ideal

The reconstruction of a three-dimensional image of a specimen from a set of electron micrographs reduces, under certain assumptions about the imaging process in the microscope, to the mathematical problem of reconstructing a density distribution from a set of its plane projections.In the absence of noise we can formulate a purely geometrical criterion, which, for a general object, fixes the resolution attainable from a given finite number of views in terms of the size of the object. For simplicity we take the ideal case of projections collected by a series of m equally spaced tilts about a single axis.

X-Ray Analysis of Frozen Hydrated Sections:The Unconventional Approach

1982 ◽  
Vol 40 ◽  
pp. 754-757
Author(s):  
R. Beeuwkes ◽  
A. Saubermann ◽  
P. Echlin ◽  
S. Churchill
Keyword(s):  
Ratio Method ◽  
Frozen Sections ◽  
Technical Problems ◽  
Sample Handling ◽  
X Ray ◽  
Common Group ◽  
Ideal Method ◽  
Classic Paper ◽  
Physical Principles ◽  
The Ideal

Fifteen years ago, Hall described clearly the advantages of the thin section approach to biological x-ray microanalysis, and described clearly the ratio method for quantitive analysis in such preparations. In this now classic paper, he also made it clear that the ideal method of sample preparation would involve only freezing and sectioning at low temperature. Subsequently, Hall and his coworkers, as well as others, have applied themselves to the task of direct x-ray microanalysis of frozen sections. To achieve this goal, different methodological approachs have been developed as different groups sought solutions to a common group of technical problems. This report describes some of these problems and indicates the specific approaches and procedures developed by our group in order to overcome them. We acknowledge that the techniques evolved by our group are quite different from earlier approaches to cryomicrotomy and sample handling, hence the title of our paper. However, such departures from tradition have been based upon our attempt to apply basic physical principles to the processes involved. We feel we have demonstrated that such a break with tradition has valuable consequences.

Sign in / Sign up

Export Citation Format

Share Document