Semigroups whose right ideals are finitely generated

Finitely Generated ◽

Equivalent Formulation ◽

Regular Semigroups ◽

Strong Semilattice ◽

Ascending Chain Condition ◽

Completely Simple

Abstract We call a semigroup $S$ weakly right noetherian if every right ideal of $S$ is finitely generated; equivalently, $S$ satisfies the ascending chain condition on right ideals. We provide an equivalent formulation of the property of being weakly right noetherian in terms of principal right ideals, and we also characterize weakly right noetherian monoids in terms of their acts. We investigate the behaviour of the property of being weakly right noetherian under quotients, subsemigroups and various semigroup-theoretic constructions. In particular, we find necessary and sufficient conditions for the direct product of two semigroups to be weakly right noetherian. We characterize weakly right noetherian regular semigroups in terms of their idempotents. We also find necessary and sufficient conditions for a strong semilattice of completely simple semigroups to be weakly right noetherian. Finally, we prove that a commutative semigroup $S$ with finitely many archimedean components is weakly (right) noetherian if and only if $S/\mathcal {H}$ is finitely generated.

Eventually Pointed Principally Ordered Regular Semigroups

Sultan Qaboos University Journal for Science [SQUJS] ◽

10.24200/squjs.vol24iss2pp139-146 ◽

2020 ◽

Vol 24 (2) ◽

pp. 139

Author(s):

G.A. Pinto

Keyword(s):

Positive Integer ◽

Regular Semigroup ◽

Sufficient Conditions ◽

Regular Semigroups ◽

Completely Simple

An ordered regular semigroup, , is said to be principally ordered if for every there exists . A principally ordered regular semigroup is pointed if for every element, we have . Here we investigate those principally ordered regular semigroups that are eventually pointed in the sense that for all there exists a positive integer, , such that . Necessary and sufficient conditions for an eventually pointed principally ordered regular semigroup to be naturally ordered and to be completely simple are obtained. We describe the subalgebra of generated by a pair of comparable idempotents and such that .

Eventually regular semigroups that are group-bound

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700004573 ◽

1986 ◽

Vol 34 (1) ◽

pp. 127-132 ◽

Cited By ~ 5

Author(s):

P. M. Edwards

Keyword(s):

Sufficient Conditions ◽

Regular Semigroups ◽

Necessary and sufficient conditions are given for certain classes of eventually regular semigroups to the group-bound or even periodic.

Inertial subalgebras of algebras possessing finite automorphism groups

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700012295 ◽

1979 ◽

Vol 28 (3) ◽

pp. 335-345 ◽

Author(s):

Nicholas S. Ford

Keyword(s):

Finite Group ◽

Commutative Ring ◽

Jacobson Radical ◽

Automorphism Groups ◽

Sufficient Conditions ◽

Finitely Generated ◽

Fixed Ring ◽

A Finite Group ◽

AbstractLet R be a commutative ring with identity, and let A be a finitely generated R-algebra with Jacobson radical N and center C. An R-inertial subalgebra of A is a R-separable subalgebra B with the property that B+N=A. Suppose A is separable over C and possesses a finite group G of R-automorphisms whose restriction to C is faithful with fixed ring R. If R is an inertial subalgebra of C, necessary and sufficient conditions for the existence of an R-inertial subalgebra of A are found when the order of G is a unit in R. Under these conditions, an R-inertial subalgebra B of A is characterized as being the fixed subring of a group of R-automorphisms of A. Moreover, A ⋍ B ⊗R C. Analogous results are obtained when C has an R-inertial subalgebra S ⊃ R.

On the equivalence of cancellative extensions of commutative cancellative semigroups by groups

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700009459 ◽

1971 ◽

Vol 12 (2) ◽

pp. 187-192

Author(s):

Charles V. Heuer

Keyword(s):

Abelian Group ◽

Cyclic Group ◽

Sufficient Conditions ◽

Ideal Extension ◽

Cancellative Semigroup ◽

Finitely Generated ◽

In [1] D. W. Miller and the author established necessary and sufficient conditions for the existence of a cancellative (ideal) extension of a commutative cancellative semigroup by a cyclic group with zero. The purpose of this paper is to extend these results to cancellative extensions by any finitely generated Abelian group with zero and to establish in this general case conditions under which two such extensions are equivalent.

Isomorphisms of Prime Goldie Semi-Principal Left Ideal Rings, II

Canadian Mathematical Bulletin ◽

10.4153/cmb-1988-053-3 ◽

1988 ◽

Vol 31 (3) ◽

pp. 374-379 ◽

Author(s):

Kenneth G. Wolfson

Keyword(s):

Left Ideal ◽

Endomorphism Ring ◽

Finite Rank ◽

Sufficient Conditions ◽

Free Module ◽

Finitely Generated ◽

Ore Domain ◽

AbstractA prime Goldie ring K, in which each finitely generated left ideal is principal is the endomorphism ring E(F, A) of a free module A, of finite rank, over an Ore domain F. We determine necessary and sufficient conditions to insure that whenever K ≅ E(F, A) ≅ E(G, B) (with A and B free and finitely generated over domains F and G) then (F, A) is semi-linearly isomorphic to (G, B). We also show, by example, that it is possible for K ≅ E(F, A ) ≅ E(G, B), with F and G, not isomorphic.

Hilbert rings and G(oldman)-rings issued from amalgamated algebras

Journal of Algebra and Its Applications ◽

10.1142/s0219498818500238 ◽

2018 ◽

Vol 17 (02) ◽

pp. 1850023 ◽

Author(s):

L. Izelgue ◽

O. Ouzzaouit

Keyword(s):

New York ◽

Commutative Algebra ◽

Quotient Ring ◽

Sufficient Conditions ◽

Ring Homomorphism ◽

Local Rings ◽

Finitely Generated ◽

The Way

Let [Formula: see text] and [Formula: see text] be two rings, [Formula: see text] an ideal of [Formula: see text] and [Formula: see text] be a ring homomorphism. The ring [Formula: see text] is called the amalgamation of [Formula: see text] with [Formula: see text] along [Formula: see text] with respect to [Formula: see text]. It was proposed by D’anna and Fontana [Amalgamated algebras along an ideal, Commutative Algebra and Applications (W. de Gruyter Publisher, Berlin, 2009), pp. 155–172], as an extension for the Nagata’s idealization, which was originally introduced in [Nagata, Local Rings (Interscience, New York, 1962)]. In this paper, we establish necessary and sufficient conditions under which [Formula: see text], and some related constructions, is either a Hilbert ring, a [Formula: see text]-domain or a [Formula: see text]-ring in the sense of Adams [Rings with a finitely generated total quotient ring, Canad. Math. Bull. 17(1) (1974)]. By the way, we investigate the transfer of the [Formula: see text]-property among pairs of domains sharing an ideal. Our results provide original illustrating examples.

A classification of vertex-transitive Cayley digraphs of strong semilattices of completely simple semigroups

Mathematica Slovaca ◽

10.2478/s12175-012-0048-3 ◽

2012 ◽

Vol 62 (5) ◽

Author(s):

Shou-feng Wang ◽

Di Zhang

Keyword(s):

Sufficient Conditions ◽

Completely Simple Semigroups ◽

Vertex Transitive ◽

Completely Simple

AbstractWith the help of a property of completely simple semigroups proved in this paper we give necessary and sufficient conditions for vertex-transitivity of Cayley digraphs of strong semilattices of completely simple semigroups.

A NEW CONSTRUCTION FOR REGULAR SEMIGROUPS WITH QUASI-IDEAL ORTHODOX TRANSVERSALS

Journal of the Australian Mathematical Society ◽

10.1017/s1446788708000566 ◽

2009 ◽

Vol 86 (2) ◽

pp. 177-187 ◽

Cited By ~ 4

Author(s):

XIANGJUN KONG ◽

XIANZHONG ZHAO

Keyword(s):

Regular Semigroup ◽

Structure Theorem ◽

Sufficient Conditions ◽

Green’S Relations ◽

Regular Semigroups ◽

New Construction ◽

Equivalent Conditions ◽

Orthodox Semigroups

AbstractIn any regular semigroup with an orthodox transversal, we define two sets R and L using Green’s relations and give necessary and sufficient conditions for them to be subsemigroups. By using R and L, some equivalent conditions for an orthodox transversal to be a quasi-ideal are obtained. Finally, we give a structure theorem for regular semigroups with quasi-ideal orthodox transversals by two orthodox semigroups R and L.

EXCHANGE RINGS OVER WHICH EVERY REGULAR MATRIX ADMITS DIAGONAL REDUCTION

Journal of Algebra and Its Applications ◽

10.1142/s0219498804000770 ◽

2004 ◽

Vol 03 (02) ◽

pp. 207-217 ◽

Cited By ~ 5

Author(s):

HUANYIN CHEN

Keyword(s):

Sufficient Conditions ◽

Exchange Ring ◽

Regular Matrix ◽

Finitely Generated ◽

Exchange Rings ◽

Diagonal Reduction

In this paper, we investigate the necessary and sufficient conditions on exchange rings, under which every regular matrix admits diagonal reduction. Also we show that an exchange ring R is strongly separative if and only if for any finitely generated projective right R-module C, if A and B are any right R-modules such that 2C⊕A≅C⊕B, then C⊕A≅B.

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.