scholarly journals Rings generated by their regular elements

1984 ◽  
Vol 25 (1) ◽  
pp. 1-5 ◽  
Author(s):  
A. W. Chatters ◽  
S. M. Ginn
Keyword(s):  
Direct Summand ◽  
Regular Element ◽  
Prime Factor ◽  
Artinian Ring ◽  
Noetherian Rings ◽  
Multiplicative Property ◽  
Regular Elements ◽  
Ring Of Integers ◽  
Factor Ring ◽  
Zero Divisors

The units of a ring R are defined by means of a multiplicative property, but in many cases they generate R additively. For example, it is shown in [5, Proposition 6] that if R is a semi-simple Artinian ring then every element of R is a sum of units if and only if the ring S = ℤ/2ℤ⊕ℤ/2ℤ is not a direct summand of R, where ℤ denotes the ring of integers. The theme of this paper is to investigate the corresponding situation concerning regular elements, i. e. elements which are not zero-divisors. We show that if R is a semi-prime right Goldie ring then every element of R is a sum of regular elements if and only if R does not have the ring S defined above as a direct summand (Corollary 2.9). We also characterise those Noetherian rings R such that every element of R is a sum of regular elements (Theorem 2. 6). The characterisation is in terms of the nature of certain prime factor rings of R, and it is again the presence of the ring S, this time in a particular way as a factor ring of R, which prevents R from being generated by its regular elements. If R has no non-zero Artinian one-sided ideals or if 2 is a regular element of R, then every element of R is a sum of regular elements (Corollaries 2. 5 and 2. 7). As an application we show in Section 3 that, for many Noetherian rings R, the set of elements of R which are divisible by every regular element of R is a two-sided ideal of R.

scholarly journals A note on Noetherian orders in Artinian rings

1979 ◽  
Vol 20 (2) ◽  
pp. 125-128 ◽  
Author(s):  
A. W. Chatters
Keyword(s):  
Noetherian Ring ◽  
Quotient Ring ◽  
Triangular Matrix ◽  
Idempotent Element ◽  
Noetherian Rings ◽  
Central Idempotent ◽  
Matrix Rings ◽  
Regular Elements ◽  
Zero Divisors ◽  
Left And Right

Throughout this note, rings are associative with identity element but are not necessarily commutative. Let R be a left and right Noetherian ring which has an Artinian (classical) quotient ring. It was shown by S. M. Ginn and P. B. Moss [2, Theorem 10] that there is a central idempotent element e of R such that eR is the largest Artinian ideal of R. We shall extend this result, using a different method of proof, to show that the idempotent e is also related to the socle of R/N (where N, throughout, denotes the largest nilpotent ideal of R) and to the intersection of all the principal right (or left) ideals of R generated by regular elements (i.e. by elements which are not zero-divisors). There are many examples of left and right Noetherian rings with Artinian quotient rings, e.g. commutative Noetherian rings in which all the associated primes of zero are minimal together with full or triangular matrix rings over such rings. It was shown by L. W. Small that if R is any left and right Noetherian ring then R has an Artinian quotient ring if and only if the regular elements of R are precisely the elements c of R such that c + N is a regular element of R/N (for further details and examples see [5] and [6]). By the largest Artinian ideal of R we mean the sum of all the Artinian right ideals of R, and it was shown by T. H. Lenagan in [3] that this coincides in any left and right Noetherian ring R with the sum of all the Artinian left ideals of R.

Inverse complements and strongly unit regular elements

2021 ◽  
pp. 2250190
Author(s):  
Umashankara Kelathaya ◽  
Savitha Varkady ◽  
Manjunatha Prasad Karantha
Keyword(s):  
Partial Order ◽  
Regular Element ◽  
Generalized Inverse ◽  
Generalized Inverses ◽  
Order Unit ◽  
Regular Elements ◽  
Strongly Regular ◽  
Minus Partial Order

In this paper, the notion of “strongly unit regular element”, for which every reflexive generalized inverse is associated with an inverse complement, is introduced. Noting that every strongly unit regular element is unit regular, some characterizations of unit regular elements are obtained in terms of inverse complements and with the help of minus partial order. Unit generalized inverses of given unit regular element are characterized as sum of reflexive generalized inverses and the generators of its annihilators. Surprisingly, it has been observed that the class of strongly regular elements and unit regular elements are the same. Also, several classes of generalized inverses are characterized in terms of inverse complements.

scholarly journals On rings all of whose factor rings are integral domains

1993 ◽  
Vol 55 (3) ◽  
pp. 325-333
Author(s):  
Yasuyuki Hirano
Keyword(s):  
Positive Integer ◽  
Zero Divisor ◽  
Arbitrary Positive Integer ◽  
Integral Domains ◽  
Factor Ring ◽  
Zero Divisors ◽  
Proper Quotient

AbstractA ring R is called a (proper) quotient no-zero-divisor ring if every (proper) nonzero factor ring of R has no zero-divisors. A characterization of a quotient no-zero-divisor ring is given. Using it, the additive groups of quotient no-zero-divisor rings are determined. In addition, for an arbitrary positive integer n, a quotient no-zero-divisor ring with exactly n proper ideals is constructed. Finally, proper quotient no-zero-divisor rings and their additive groups are classified.

scholarly journals Strongly regular elements of noetherian rings

1984 ◽  
Vol 91 (2) ◽  
pp. 410-429 ◽  
Author(s):  
Alfred Goldie ◽  
Günter Krause
Keyword(s):  
Noetherian Rings ◽  
Regular Elements ◽  
Strongly Regular

scholarly journals The Centralizer of an I-Matrix in M2(R/I), R a UFD

2014 ◽  
Vol 21 (04) ◽  
pp. 615-626
Author(s):  
Magdaleen S. Marais
Keyword(s):  
Matrix Ring ◽  
Nonzero Ideal ◽  
Factor Ring ◽  
Zero Divisors ◽  
Matrix Formula

The concept of an I-matrix in the full 2 × 2 matrix ring M2(R/I), where R is an arbitrary UFD and I is a nonzero ideal in R, is introduced. We obtain a concrete description of the centralizer of an I-matrix [Formula: see text] in M2(R/I) as the sum of two subrings 𝒮1 and 𝒮2 of M2(R/I), where 𝒮1 is the image (under the natural epimorphism from M2(R) to M2(R/I)) of the centralizer in M2(R) of a pre-image of [Formula: see text], and the entries in 𝒮2 are intersections of certain annihilators of elements arising from the entries of [Formula: see text]. It turns out that if R is a PID, then every matrix in M2(R/I) is an I-matrix. However, this is not the case if R is a UFD in general. Moreover, for every factor ring R/I with zero divisors and every n ≥ 3, there is a matrix for which the mentioned concrete description is not valid.

A note on r-precious ring

2020 ◽  
pp. 2150231
Author(s):  
Nitin Bisht
Keyword(s):  
Regular Element ◽  
Nilpotent Element ◽  
Idempotent Element ◽  
Von Neumann ◽  
Regular Elements ◽  
Basic Properties ◽  
Euclidean Domain ◽  
Von Neumann Regular

An element of a ring [Formula: see text] is said to be [Formula: see text]-precious if it can be written as the sum of a von Neumann regular element, an idempotent element and a nilpotent element. If all the elements of a ring [Formula: see text] are [Formula: see text]-precious, then [Formula: see text] is called an [Formula: see text]-precious ring. We study some basic properties of [Formula: see text]-precious rings. We also characterize von Neumann regular elements in [Formula: see text] when [Formula: see text] is a Euclidean domain and by this argument, we produce elements that are [Formula: see text]-precious but either not [Formula: see text]-clean or not precious.

scholarly journals Maximal inverse subsemigroups of S(X)

1983 ◽  
Vol 24 (1) ◽  
pp. 53-64 ◽  
Author(s):  
Bridget B. Baird
Keyword(s):  
Topological Space ◽  
Inverse Semigroup ◽  
Regular Element ◽  
Continuous Functions ◽  
Regular Elements ◽  
Inverse Subsemigroups

If X is a topological space then S(X) will denote the semigroup, under composition, of all continuous functions from X into X. An element f in a semigroup is regular if there is an element g such that fgf = f. The regular elements of S(X) will be denoted by R(X). Elements f and g are inverses of each other if fgf = f and gfg = g. Every regular element has an inverse [1]. If every element in a semigroup has a unique inverse then the semigroup is an inverse semigroup. In this paper we examine maximal inverse subsemigroups of S(X).

ℎ-local Rings

2021 ◽  
pp. 93-109
Author(s):  
L. Klingler ◽  
A. Omairi
Keyword(s):  
Local Ring ◽  
Nonzero Element ◽  
Local Rings ◽  
Local Domain ◽  
Content Type ◽  
Regular Elements ◽  
Zero Divisors ◽  
Maximal Ideals ◽  
Regular Ideal ◽  
Definition Of

In the 1960’s, Matlis defined an h h -local domain to be a (commutative) integral domain in which each nonzero element is contained in only finitely many maximal ideals and each nonzero prime ideal is contained in a unique maximal ideal. For rings with zero-divisors, by changing “nonzero” to “regular,” one obtains the definition of an h h -local ring. Nearly two dozen equivalent characterizations of h h -local domain have appeared in the literature. We show that most of these remain equivalent to h h -local ring if one also replaces “localization” by “regular localization” and assumes that the ring is a Marot ring (i.e., every regular ideal is generated by its regular elements).

scholarly journals The Ohm–Rush content function II. Noetherian rings, valuation domains, and base change

2019 ◽  
Vol 18 (05) ◽  
pp. 1950100
Author(s):  
Neil Epstein ◽  
Jay Shapiro
Keyword(s):  
Noetherian Ring ◽  
Artinian Ring ◽  
Base Change ◽  
Noetherian Rings ◽  
Prime Characteristic ◽  
Field Extensions ◽  
Dimension Formula ◽  
Valuation Domains ◽  
Prüfer Domains ◽  
Polynomial Extensions

The notion of an Ohm–Rush algebra, and its associated content map, has connections with prime characteristic algebra, polynomial extensions, and the Ananyan–Hochster proof of Stillman’s conjecture. As further restrictions are placed (creating the increasingly more specialized notions of weak content, semicontent, content, and Gaussian algebras), the construction becomes more powerful. Here we settle the question in the affirmative over a Noetherian ring from [N. Epstein and J. Shapiro, The Ohm-Rush content function, J. Algebra Appl. 15(1) (2016) 1650009, 14 pp.] of whether a faithfully flat weak content algebra is semicontent (and over an Artinian ring of whether such an algebra is content), though both questions remain open in general. We show that in content algebra maps over Prüfer domains, heights are preserved and a dimension formula is satisfied. We show that an inclusion of nontrivial valuation domains is a content algebra if and only if the induced map on value groups is an isomorphism, and that such a map induces a homeomorphism on prime spectra. Examples are given throughout, including results that show the subtle role played by properties of transcendental field extensions.

On coseparable and 𝔪-coseparable modules

2020 ◽  
pp. 2150031
Author(s):  
Rachid Ech-chaouy ◽  
Abdelouahab Idelhadj ◽  
Rachid Tribak
Keyword(s):  
Direct Summand ◽  
Noetherian Rings ◽  
Direct Products ◽  
Finitely Generated ◽  
Direct Sums ◽  
Direct Sum ◽  
Free Modules

A module [Formula: see text] is called coseparable ([Formula: see text]-coseparable) if for every submodule [Formula: see text] of [Formula: see text] such that [Formula: see text] is finitely generated ([Formula: see text] is simple), there exists a direct summand [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] is finitely generated. In this paper, we show that free modules are coseparable. We also investigate whether or not the ([Formula: see text]-)coseparability is stable under taking submodules, factor modules, direct summands, direct sums and direct products. We show that a finite direct sum of coseparable modules is not, in general, coseparable. But the class of [Formula: see text]-coseparable modules is closed under finite direct sums. Moreover, it is shown that the class of coseparable modules over noetherian rings is closed under finite direct sums. A characterization of coseparable modules over noetherian rings is provided. It is also shown that every lifting (H-supplemented) module is coseparable ([Formula: see text]-coseparable).

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