On Regular Elements in an Incline

Generalized Inverses ◽

Regular Elements ◽

Idempotent Semirings ◽

Inclines are additively idempotent semirings in which products are less than (or) equal to either factor. Necessary and sufficient conditions for an element in an incline to be regular are obtained. It is proved that every regular incline is a distributive lattice. The existence of the Moore-Penrose inverse of an element in an incline with involution is discussed. Characterizations of the set of all generalized inverses are presented as a generalization and development of regular elements in a∗-regular ring.

Subdirect products of semirings

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171201003696 ◽

2001 ◽

Vol 26 (9) ◽

pp. 539-545

Author(s):

P. Mukhopadhyay

Keyword(s):

Distributive Lattice ◽

Subdirect Product ◽

Sufficient Conditions ◽

Subdirect Products ◽

Bandelt and Petrich (1982) proved that an inversive semiringSis a subdirect product of a distributive lattice and a ring if and only ifSsatisfies certain conditions. The aim of this paper is to obtain a generalized version of this result. The main purpose of this paper however, is to investigate, what new necessary and sufficient conditions need we impose on an inversive semiring, so that, in its aforesaid representation as a subdirect product, the ring involved can be gradually enriched to a field. Finally, we provide a construction of fullE-inversive semirings, which are subdirect products of a semilattice and a ring.

Semicritical Rings and the Quotient Problem

Canadian Mathematical Bulletin ◽

10.4153/cmb-1984-025-7 ◽

1984 ◽

Vol 27 (2) ◽

pp. 160-170

Author(s):

Karl A. Kosler

Keyword(s):

Quotient Ring ◽

Sufficient Conditions ◽

Prime Ideals ◽

Regular Elements ◽

The Right ◽

Minimal Prime ◽

Necessary And Sufficient ◽

Quotient Rings ◽

The Relationship

AbstractThe purpose of this paper is to examine the relationship between the quotient problem for right noetherian nonsingular rings and the quotient problem for semicritical rings. It is shown that a right noetherian nonsingular ring R has an artinian classical quotient ring iff certain semicritical factor rings R/Ki, i = 1,…,n, possess artinian classical quotient rings and regular elements in R/Ki lift to regular elements of R for all i. If R is a two sided noetherian nonsingular ring, then the existence of an artinian classical quotient ring is equivalent to each R/Ki possessing an artinian classical quotient ring and the right Krull primes of R consisting of minimal prime ideals. If R is also weakly right ideal invariant, then the former condition is redundant. Necessary and sufficient conditions are found for a nonsingular semicritical ring to have an artinian classical quotient ring.

Regular, Commutative, Maximal Semigroups of Quotients

Canadian Mathematical Bulletin ◽

10.4153/cmb-1975-018-3 ◽

1975 ◽

Vol 18 (1) ◽

pp. 99-104 ◽

Cited By ~ 3

Author(s):

Jurgen Rompke

Keyword(s):

Commutative Semigroup ◽

Regular Ring ◽

Sufficient Conditions ◽

Commutative Rings ◽

Natural Question ◽

Commutative Case ◽

Ring Of Quotients ◽

Singular Ideal ◽

A well-known theorem which goes back to R. E. Johnson [4], asserts that if R is a ring then Q(R), its maximal ring of quotients is regular (in the sense of v. Neumann) if and only if the singular ideal of R vanishes. In the theory of semigroups a natural question is therefore the following: Do there exist properties which characterize those semigroups whose maximal semigroups of quotients are regular? Partial answers to this question have been given in [3], [7] and [8]. In this paper we completely solve the commutative case, i.e. we give necessary and sufficient conditions for a commutative semigroup S in order that Q(S), the maximal semigroup of quotients, is regular. These conditions reflect very closely the property of being semiprime, which in the theory of commutative rings characterizes those rings which have a regular ring of quotients.

Further results on generalized inverses in rings with involution

Electronic Journal of Linear Algebra ◽

10.13001/1081-3810.2958 ◽

2015 ◽

Vol 30 ◽

Cited By ~ 10

Author(s):

N. Castro-Gonzalez ◽

Jianlong Chen ◽

Long Wang

Keyword(s):

Sufficient Conditions ◽

Generalized Inverses ◽

Matrix Product ◽

Explicit Formulas ◽

Block Matrices ◽

Unital Ring ◽

Rings With Involution ◽

Let R be a unital ring with an involution. Necessary and sufficient conditions for the existence of the Bott-Duffin inverse of a in R relative to a pair of self-adjoint idempotents (e, f) are derived. The existence of a {1, 3}-inverse, {1, 4}-inverse, and the Moore-Penrose inverse of a matrix product is characterized, and explicit formulas for their computations are obtained. Some applications to block matrices over a ring are given.

Ideals on neutrosophic extended triplet groups

AIMS Mathematics ◽

10.3934/math.2022264 ◽

2021 ◽

Vol 7 (3) ◽

pp. 4767-4777

Author(s):

Xin Zhou ◽

◽

Xiao Long Xin ◽

Keyword(s):

Prime Ideal ◽

Distributive Lattice ◽

Topological Structure ◽

Hausdorff Space ◽

Sufficient Conditions ◽

Prime Ideals ◽

Ideal Space ◽

<abstract><p>In this paper, we introduce the concept of (prime) ideals on neutrosophic extended triplet groups (NETGs) and investigate some related properties of them. Firstly, we give characterizations of ideals generated by some subsets, which lead to a construction of a NETG by endowing the set consisting of all ideals with a special multiplication. In addition, we show that the set consisting of all ideals is a distributive lattice. Finally, by introducing the topological structure on the set of all prime ideals on NETGs, we obtain the necessary and sufficient conditions for the prime ideal space to become a $ T_{1} $-space and a Hausdorff space. </p></abstract>

Some additive and multiplicative results for generalized inverses

Filomat ◽

10.2298/fil1509049n ◽

2015 ◽

Vol 29 (9) ◽

pp. 2049-2057

Author(s):

Jovana Nikolov-Radenkovic

Keyword(s):

Hilbert Space ◽

Sufficient Conditions ◽

Generalized Inverses ◽

Reverse Order ◽

Reverse Order Law ◽

Necessary And Sufficient ◽

Regular Operators

In this paper we give necessary and sufficient conditions for A1{1,3} + A2{1, 3}+ ... + Ak{1,3} ? (A1 + A2 + ... + Ak){1,3} and A1{1,4} + A2{1,4} + ... + Ak{1,4} ? (A1 + A2 + ... + Ak){1,4} for regular operators on Hilbert space. We also consider similar inclusions for {1,2,3}- and {1,2,4}-i inverses. We give some new results concerning the reverse order law for reflexive generalized inverses.

FREE TERNARY ALGEBRAS

International Journal of Algebra and Computation ◽

10.1142/s0218196700000340 ◽

2000 ◽

Vol 10 (06) ◽

pp. 739-749 ◽

Cited By ~ 3

Author(s):

RAYMOND BALBES

Keyword(s):

Distributive Lattice ◽

Sufficient Conditions ◽

Free Generator ◽

Finitely Generated ◽

Bounded Distributive Lattice ◽

Ternary Algebra ◽

Free Generators ◽

Ternary Algebras ◽

A ternary algebra is a bounded distributive lattice with additonal operations e and ~ that satisfies (a+b)~=a~b~, a~~=a, e≤a+a~, e~= e and 0~=1. This article characterizes free ternary algebras by giving necessary and sufficient conditions on a set X of free generators of a ternary algebra L, so that X freely generates L. With this characterization, the free ternary algebra on one free generator is displayed. The poset of join irreducibles of finitely generated free ternary algebras is characterized. The uniqueness of the set of free generators and their pseudocomplements is also established.

Properties of Stone almost distributive lattices

Asian-European Journal of Mathematics ◽

10.1142/s1793557115500114 ◽

2015 ◽

Vol 08 (01) ◽

pp. 1550011

Author(s):

G. C. Rao ◽

Mihret Alamneh

Keyword(s):

Distributive Lattice ◽

Sufficient Conditions ◽

Distributive Lattices ◽

Basic Facts ◽

In this paper, we study different properties of Stone almost distributive lattice (Stone ADL), prove basic facts on a Stone ADL and derive a necessary and sufficient conditions for an ADL L to be a Stone ADL.

PRIME IDEAL CHARACTERIZATION OF STONE ADLs

Asian-European Journal of Mathematics ◽

10.1142/s179355711000026x ◽

2010 ◽

Vol 03 (02) ◽

pp. 357-367 ◽

Cited By ~ 1

Author(s):

U. M. Swamy ◽

S. Ramesh ◽

Ch. Shanthi Sundar Raj

Keyword(s):

Prime Ideal ◽

Distributive Lattice ◽

Sufficient Conditions ◽

In this paper we obtain certain necessary and sufficient conditions for an almost distributive lattice to become a Stone almost distributive lattice in topological and algebraic terms.

BL-ALMOST DISTRIBUTIVE LATTICES

Asian-European Journal of Mathematics ◽

10.1142/s1793557112500222 ◽

2012 ◽

Vol 05 (02) ◽

pp. 1250022 ◽

Cited By ~ 1

Author(s):

G. C. Rao ◽

Naveen Kumar Kakumanu

Keyword(s):

Distributive Lattice ◽

Sufficient Conditions ◽

Distributive Lattices ◽

The concept of BL-almost distributive lattice (BL-ADL) is introduced and necessary and sufficient conditions for an ADL to become a BL-ADL are derived. Different characterizations of a BL-ADL are obtained.