Direct Product Decomposition of Commutative Semisimple Rings

Abian, Alexander

doi:10.2307/2037396

Cited by 19 publications

(29 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For abelian PP rings our order coincides with a relation introduced by Sussman [25], Abian [1,2] and further studied by Chacron [7]. Burmistrovic [6] investigated Sussman's order on separative semigroups.…”

supporting

confidence: 67%

“…Conversely, suppose that R a^Rb then, aebS 1 . But then a = bx for some element x belonging to S 1 .…”

Section: Proof Let X and Y Be The Inverses Of A And B Respectively mentioning

confidence: 99%

“…Now, fa = b and fS 1 = {fz,b} and aS l = {a,z) and so fS l r>aS l = {z). But faS l =bS l = {z,b) which means that fS l naS l ^faS 1 . So S is not type A, which implies that S is not IC.…”

Section: Locally Type a Semigroupsmentioning

confidence: 99%

See 2 more Smart Citations

The natural partial order on an abundant semigroup

Lawson

1987

Proceedings of the Edinburgh Mathematical Society

View full text Add to dashboard Cite

In this paper we will study the properties of a natural partial order which may be defined on an arbitrary abundant semigroup: in the case of regular semigroups we recapture the order introduced by Nambooripad [24]. For abelian PP rings our order coincides with a relation introduced by Sussman [25], Abian [1,2] and further studied by Chacron [7]. Burmistrovic [6] investigated Sussman's order on separative semigroups. In the abundant case his order coincides with ours: some order theoretic properties of such semigroups may be found in a paper by Burgess [5].Many properties and constructions on abundant semigroups may be described in terms of its natural partial order: one of the main results of Section 2 is the connection we establish between idempotent connectedness and the partial order being, in some sense, self-dual. In Section 3 we extend Nambooripad's Theorem 3.3 [24] and show that the order is compatible with the multiplication on a concordant semigroup just when the semigroup is locally type A. In Section 4 we obtain a description of the finest 0-restricted primitive good congruence on a concordant semigroup. Section 1 is a preliminary section in which we consider, in particular, the behaviour of good homomorphisms between abundant semigroups and obtain a generalisation of Lallement's Lemma. PreliminariesWe will assume some familiarity with the contents of [11] and [15]: the only divergence from the terminology established there is that we prefer to call *-ideals good ideals, bringing them in line with the good homomorphisms.The class of idempotent connected (IC) abundant semigroups was introduced by El-Qallali and Fountain [11]. We begin this section with an alternative characterisation of these semigroups due to Fountain, which, for the purposes of this paper, may serve as an alternative definition.

show abstract

supporting

confidence: 67%

“…Conversely, suppose that R a^Rb then, aebS 1 . But then a = bx for some element x belonging to S 1 .…”

Section: Proof Let X and Y Be The Inverses Of A And B Respectively mentioning

confidence: 99%

See 1 more Smart Citation

The natural partial order on an abundant semigroup

Lawson

1987

Proceedings of the Edinburgh Mathematical Society

View full text Add to dashboard Cite

show abstract

“…If R is orthogonally complete, (1) shows that each R q will be orthogonally complete. But when R q is a ring it is regular so, when this occurs, if R is orthogonally complete then R q is self-injective.…”

Section: Corollary 6 (Renault [13]) a Strongly Regular Ring Is Leftmentioning

confidence: 99%

“…EXAMPLE 13. Let R be the ring of all continuous real-valued functions on [0,1]. It will be shown that R does not have an orthogonal completion.…”

Section: Let R Be Reduced and Such That Ar F]br = 0 Implies Ab = 0 Tmentioning

confidence: 99%