scholarly journals Cohen Macaulay Bipartite Graphs and Regular Element on the Powers of Bipartite Edge Ideals

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 762
Author(s):  
Arindam Banerjee ◽  
Vivek Mukundan
Keyword(s):  
Regular Element ◽  
Bipartite Graphs ◽  
Edge Ideals ◽  
Regular Elements

In this article, we discuss new characterizations of Cohen-Macaulay bipartite edge ideals. For arbitrary bipartite edge ideals I ( G ) , we also discuss methods to recognize regular elements on I ( G ) s for all s ≥ 1 in terms of the combinatorics of the graph G.

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 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.

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).

A CLOSURE OPERATION IN RINGS

2001 ◽  
Vol 12 (07) ◽  
pp. 791-812
Author(s):  
PERE ARA ◽  
GERT K. PEDERSEN ◽  
FRANCESC PERERA
Keyword(s):  
Regular Element ◽  
Closure Operation ◽  
Von Neumann ◽  
Regular Elements ◽  
C Algebras ◽  
Partial Inverse ◽  
Regular Ideal ◽  
Von Neumann Regular ◽  
Invertible Elements ◽  
Maximal Regular Ideal

We study the operation E → cl (E) defined on subsets E of a unital ring R, where x ∈ cl (E) if (x + Rb) ∩ E ≠ ∅ for each b in R such that Rx + Rb = R. This operation, which strongly resembles a closure, originates in algebraic K-theory. For any left ideal L we show that cl (L) equals the intersection of the maximal left ideals of R containing L. Moreover, cl (Re) = Re + rad (R) if e is an idempotent in R, and cl (I) = I for a two-sided ideal I precisely when I is semi-primitive in R (i.e. rad (R/I) = 0). We then explore a special class of von Neumann regular elements in R, called persistently regular and characterized by forming an "open" subset Rpr in R, i.e. cl (R\Rpr) = R\Rpr. In fact, R\Rpr = cl (R\Rr), so that Rpr is the "algebraic interior" of the set Rr of regular elements. We show that a regular element x with partial inverse y is persistently regular, if and only if the skew corner (1 - xy)R(1 - yx) is contained in Rr. If I reg (R) denotes the maximal regular ideal in R and [Formula: see text] the set of quasi-invertible elements, defined and studied in [4], we prove that [Formula: see text]. Specializing to C*-algebras we prove that cl (E) coincides with the norm closure of E, when E is one of the five interesting sets R-1, [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], and that Rpr coincides with the topological interior of Rr. We also show that the operation cl respects boundedness, self-adjointness and positivity.

scholarly journals TRIPLET INVARIANCE AND PARALLEL SUMS

2021 ◽  
pp. 1-9
Author(s):  
TSIU-KWEN LEE ◽  
JHENG-HUEI LIN ◽  
TRUONG CONG QUYNH
Keyword(s):  
Regular Element ◽  
Semiprime Ring ◽  
Extended Centroid ◽  
Prime Rings ◽  
Matrix Rings ◽  
Parallel Sum ◽  
Regular Elements ◽  
Regular Rings

Abstract Let R be a semiprime ring with extended centroid C and let $I(x)$ denote the set of all inner inverses of a regular element x in R. Given two regular elements $a, b$ in R, we characterise the existence of some $c\in R$ such that $I(a)+I(b)=I(c)$ . Precisely, if $a, b, a+b$ are regular elements of R and a and b are parallel summable with the parallel sum ${\cal P}(a, b)$ , then $I(a)+I(b)=I({\cal P}(a, b))$ . Conversely, if $I(a)+I(b)=I(c)$ for some $c\in R$ , then $\mathrm {E}[c]a(a+b)^{-}b$ is invariant for all $(a+b)^{-}\in I(a+b)$ , where $\mathrm {E}[c]$ is the smallest idempotent in C satisfying $c=\mathrm {E}[c]c$ . This extends earlier work of Mitra and Odell for matrix rings over a field and Hartwig for prime regular rings with unity and some recent results proved by Alahmadi et al. [‘Invariance and parallel sums’, Bull. Math. Sci.10(1) (2020), 2050001, 8 pages] concerning the parallel summability of unital prime rings and abelian regular rings.

scholarly journals Regularity 3 in edge ideals associated to bipartite graphs

2013 ◽  
Vol 39 (4) ◽  
pp. 919-937 ◽  
Author(s):  
Oscar Fernández-Ramos ◽  
Philippe Gimenez
Keyword(s):  
Bipartite Graphs ◽  
Edge Ideals

UNIT-REGULAR MODULES

2017 ◽  
Vol 60 (1) ◽  
pp. 1-15
Author(s):  
H. CHEN ◽  
W. K. NICHOLSON ◽  
Y. ZHOU
Keyword(s):  
Regular Element ◽  
Regular Ring ◽  
Stable Range ◽  
Morita Context ◽  
Regular Elements ◽  
Regular Module ◽  
Natural Extensions ◽  
Definition Of ◽  
Regular Rings ◽  
Study Unit

AbstractIn 2014, the first two authors proved an extension to modules of a theorem of Camillo and Yu that an exchange ring has stable range 1 if and only if every regular element is unit-regular. Here, we give a Morita context version of a stronger theorem. The definition of regular elements in a module goes back to Zelmanowitz in 1972, but the notion of a unit-regular element in a module is new. In this paper, we study unit-regular elements and give several characterizations of them in terms of “stable” elements and “lifting” elements. Along the way, we give natural extensions to the module case of many results about unit-regular rings. The paper concludes with a discussion of when the endomorphism ring of a unit-regular module is a unit-regular ring.

SPLIT DIALGEBRAS, SPLIT QUASI-JORDAN ALGEBRAS AND REGULAR ELEMENTS

2009 ◽  
Vol 08 (02) ◽  
pp. 191-218 ◽  
Author(s):  
RAÚL VELÁSQUEZ ◽  
RAÚL FELIPE
Keyword(s):  
Regular Element ◽  
Jordan Algebras ◽  
Regular Elements ◽  
Annihilator Ideal

The notions of annihilator ideal and split structure are studied in detail for both, dialgebras and quasi-Jordan algebras. It yields methods for additional units in the two structures. As a consequence the notion of regular element receives special attention.

A Counterexample to a Conjecture of D.F. Sanderson

1966 ◽  
Vol 9 (4) ◽  
pp. 517-517 ◽  
Author(s):  
Israel Kleiner
Keyword(s):  
Left Ideal ◽  
Regular Element ◽  
Regular Elements ◽  
Counter Example ◽  
Ring Of Quotients ◽  
Classical Ring

In [2, p. 511] Sanderson has shown that if every Large left ideal of a ring R with identity contains a regular element, and if the regular elements in R satisfy Ore's condition, then the complete (Utumi's) ring of quotients coincides with the classical ring of quotients. He conjectured that the above conditions are also necessary. The following is a counter example.

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