Reduced p.p.‐rings without identity

Xiao, Guo; Shum, Kenneth W.

doi:10.1155/ijmms/2006/92890

Cited by 3 publications

(1 citation statement)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is well known that the class of l.p.p.-rings includes the classes of (left) semihereditary rings, (left) hereditary rings, Baer rings, Baer p.q.-rings and (von Neumann) regular rings as its proper subclasses. In the literature, l.p.p.-rings and their subclasses have been extensively studied by many authors since the concept of such kind of rings was first introduced by Hattori (see, [1], [3][4][5][6][7][8][9], [12])). In this paper, we study the properties of l.p.p.-rings and left *-semisimple rings.…”

Section: Introductionmentioning

confidence: 99%

l.p.p. rings which are *-semisimple

Xiao¹,

Shum²

2007

imf

Self Cite

View full text Add to dashboard Cite

An l.p.p. ring satisfying the primitive idempotent condition is called left *-semisimple if it also satisfies the ascending condition on right *-ideals. We prove that a left *-semisimple ring can be uniquely decomposed into a direct product of a finite number of full matrix rings over certain domains, up to isomorphism. In fact, such left *-semisimple rings are generalized semisimple rings and, in particular, they are semisimple when they are finite. Mathematics Subject Classification: 16W60

show abstract

Section: Introductionmentioning

confidence: 99%

l.p.p. rings which are *-semisimple

Xiao¹,

Shum²

2007

imf

Self Cite

View full text Add to dashboard Cite

show abstract

Focal Baer semigroups and a restricted star order

Cı̄rulis¹

2019

Acta Sci. Math.

View full text Add to dashboard Cite

Semiprimeness of semigroup algebras

Guo

2021

Open Mathematics

View full text Add to dashboard Cite

Abundant semigroups originate from p.p. rings and are generalizations of regular semigroups. The main aim of this paper is to study the primeness and the primitivity of abundant semigroup algebras. We introduce and study D ∗ {{\mathcal{D}}}^{\ast } -graphs and Fountain matrices of a semigroup. Based on D ∗ {{\mathcal{D}}}^{\ast } -graphs and Fountain matrices, we determine when a contracted semigroup algebra of a primitive abundant semigroup is prime (respectively, semiprime, semiprimitive, or primitive). It is well known that for any algebra A {\mathcal{A}} with unity, A {\mathcal{A}} is primitive (prime) if and only if so is M n ( A ) {M}_{n}\left({\mathcal{A}}) . Our results can be viewed as some kind of generalizations of such a known result. In addition, it is proved that any contracted semigroup algebra of a locally ample semigroup whose set of idempotents is locally finite (respectively, locally pseudofinite and satisfying the regularity condition) is isomorphic to some contracted semigroup algebra of primitive abundant semigroups. Moreover, we obtain sufficient and necessary conditions for these classes of contracted semigroup algebras to be prime (respectively, semiprime, semiprimitive, or primitive). Finally, the structure of simple contracted semigroup algebras of idempotent-connected abundant semigroups is established. Our results enrich and extend the related results on regular semigroup algebras.

show abstract

Reduced p.p.‐rings without identity

Abstract: We give some necessary and sufficient conditions for p.p.-rings without identity to be reduced. Our results strengthen and extend the results of Fraser and Nicholson as well as some recent results we obtained on reduced p.p.-rings with identity.

Cited by 3 publications

References 12 publications

l.p.p. rings which are *-semisimple

l.p.p. rings which are *-semisimple

Focal Baer semigroups and a restricted star order

Semiprimeness of semigroup algebras

Contact Info

Product

Resources

About