Comment on: “Bi-interior ideals of semigroups”

2021 ◽  
pp. 2250056
Author(s):  
Niovi Kehayopulu
Keyword(s):  
Regular Semigroup ◽  
Left Ideal ◽  
Ideal Formula ◽  
Corrected Proof

This is about the paper “Bi-interior ideals of semigroups” by M. Murali Krishna Rao in Discuss. Math. Gen. Algebra Appl. 38 (2018) 69–78. According to Theorem 3.11 (also Theorem 3.3(8)) of the paper, the intersection of a bi-interior ideal [Formula: see text] of a semigroup [Formula: see text] and a subsemigroup [Formula: see text] of [Formula: see text] is a bi-interior ideal of [Formula: see text]. Regarding to Theorem 3.6, every bi-interior ideal of a regular semigroup is an ideal of [Formula: see text]. We give an example that the above two results are not true for semigroups. According to the same paper, if [Formula: see text] is a regular semigroup then, for every bi-interior ideal [Formula: see text], every ideal [Formula: see text] and every left ideal [Formula: see text] of [Formula: see text], we have [Formula: see text]. The proof is wrong, we provide the corrected proof. In most of the results of the paper the assumption of unity is not necessary. Care should be taken about the proofs in the paper.

A certain sublattice of the lattice of congruences on a regular semigroup

1964 ◽  
Vol 60 (3) ◽  
pp. 385-391 ◽  
Author(s):  
W. D. Munn
Keyword(s):  
Regular Semigroup ◽  
Left Ideal ◽  
Modular Lattice ◽  
Partial Ordering ◽  
Lattice Of Congruences

It is well known that, with respect to the natural partial ordering, the set of all congruences on a group forms a modular lattice. In the present paper we develop an extension of this result to the case of a regular semigroup S (α ∈ αSα for all α in S). Let Σ(ℋ) denote the set of all congruences on S with the property that congruent elements generate the same principal left ideal and the same principal right ideal of S. We show (Theorem 1) that, under the natural partial ordering, Σ(ℋ) is a modular lattice with a greatest and a least element.

Comorphic rings

2018 ◽  
Vol 17 (04) ◽  
pp. 1850075 ◽  
Author(s):  
M. Alkan ◽  
W. K. Nicholson ◽  
A. Ç. Özcan
Keyword(s):  
Left Ideal ◽  
Nilpotent Radical ◽  
Von Neumann ◽  
Ideal Formula ◽  
Frobenius Ring ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Left And Right ◽  
Preliminary Study ◽  
Regular Rings

A ring [Formula: see text] is called left comorphic if, for each [Formula: see text] there exists [Formula: see text] such that [Formula: see text] and [Formula: see text] Examples include (von Neumann) regular rings, and [Formula: see text] for a prime [Formula: see text] and [Formula: see text] One motivation for studying these rings is that the comorphic rings (left and right) are just the quasi-morphic rings, where [Formula: see text] is left quasi-morphic if, for each [Formula: see text] there exist [Formula: see text] and [Formula: see text] in [Formula: see text] such that [Formula: see text] and [Formula: see text] If [Formula: see text] here the ring is called left morphic. It is shown that [Formula: see text] is left comorphic if and only if, for any finitely generated left ideal [Formula: see text] there exists [Formula: see text] such that [Formula: see text] and [Formula: see text] Using this, we characterize when a left comorphic ring has various properties, and show that if [Formula: see text] is local with nilpotent radical, then [Formula: see text] is left comorphic if and only if it is right comorphic. We also show that a semiprime left comorphic ring [Formula: see text] is semisimple if either [Formula: see text] is left perfect or [Formula: see text] has the ACC on [Formula: see text] After a preliminary study of left comorphic rings with the ACC on [Formula: see text] we show that a quasi-Frobenius ring is left comorphic if and only if every right ideal is principal; if and only if every left ideal is a left principal annihilator. We characterize these rings as follows: The following are equivalent for a ring [Formula: see text] [Formula: see text] is quasi-Frobenius and left comorphic. [Formula: see text] is left comorphic, left perfect and right Kasch. [Formula: see text] is left comorphic, right Kasch, with the ACC on [Formula: see text] [Formula: see text] is left comorphic, left mininjective, with the ACC on [Formula: see text] Some examples of these rings are given.

Congruence relations on almost distributive lattices-II

2019 ◽  
pp. 2150005
Author(s):  
Gezahagne Mulat Addis
Keyword(s):  
Distributive Lattice ◽  
Congruence Relation ◽  
Congruence Class ◽  
Distributive Lattices ◽  
Ideal Formula ◽  
Congruence Relations

For a given ideal [Formula: see text] of an almost distributive lattice [Formula: see text], we study the smallest and the largest congruence relation on [Formula: see text] having [Formula: see text] as a congruence class.

scholarly journals Duality and the existence of weakly completely continuous elements in aB*-algebra

1972 ◽  
Vol 13 (1) ◽  
pp. 56-60 ◽  
Author(s):  
B. J. Tomiuk
Keyword(s):  
Hilbert Space ◽  
Left Ideal ◽  
Linear Operators ◽  
Completely Continuous ◽  
Identity Modulo

Ogasawara and Yoshinaga [9] have shown that aB*-algebra is weakly completely continuous (w.c.c.) if and only if it is*-isomorphic to theB*(∞)-sum of algebrasLC(HX), where eachLC(HX)is the algebra of all compact linear operators on the Hilbert spaceHx.As Kaplansky [5] has shown that aB*-algebra isB*-isomorphic to theB*(∞)-sum of algebrasLC(HX)if and only if it is dual, it follows that a5*-algebraAis w.c.c. if and only if it is dual. We have observed that, if only certain key elements of aB*-algebraAare w.c.c, thenAis already dual. This observation constitutes our main theorem which goes as follows.A B*-algebraAis dual if and only if for every maximal modular left idealMthere exists aright identity modulo M that isw.c.c.

scholarly journals Simplex algebras and their representation

1973 ◽  
Vol 14 (2) ◽  
pp. 136-144
Author(s):  
M. S. Vijayakumar
Keyword(s):  
Left Ideal ◽  
Maximal Ideal ◽  
Banach Algebras ◽  
Principal Component ◽  
Nonempty Interior ◽  
Regular Elements ◽  
Maximal Left Ideal ◽  
The Relationship ◽  
Order Identity ◽  
Real Matrices

This paper establishes a relationship (Theorem 4.1) between the approaches of A. C. Thompson [8, 9] and E. G. Effros [2] to the representation of simplex algebras, that is, real unital Banach algebras that are simplex spaces with the unit for order identity. It proves that the (nonempty) interior of the associated cone is contained in the principal component of the set of all regular elements of the algebra. It also conjectures that each maximal ideal (in the order sense—see below) of a simplex algebra contains a maximal left ideal of the algebra. This conjecture and other aspects of the relationship are illustrated by considering algebras of n × n real matrices.

On the Multiplicity and the Cohen–Macaulayness of Fiber Cones of Graded Modules

2021 ◽  
Vol 28 (01) ◽  
pp. 13-32
Author(s):  
Nguyen Tien Manh
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Primary Ideal ◽  
Graded Algebra ◽  
Finitely Generated ◽  
Ideal Formula ◽  
Noetherian Local Ring ◽  
Graded Modules ◽  
Fiber Cone

Let [Formula: see text] be a Noetherian local ring with maximal ideal [Formula: see text], [Formula: see text] an ideal of [Formula: see text], [Formula: see text] an [Formula: see text]-primary ideal of [Formula: see text], [Formula: see text] a finitely generated [Formula: see text]-module, [Formula: see text] a finitely generated standard graded algebra over [Formula: see text] and [Formula: see text] a finitely generated graded [Formula: see text]-module. We characterize the multiplicity and the Cohen–Macaulayness of the fiber cone [Formula: see text]. As an application, we obtain some results on the multiplicity and the Cohen–Macaulayness of the fiber cone[Formula: see text].

scholarly journals Comparing powers of edge ideals

2019 ◽  
Vol 18 (10) ◽  
pp. 1950184 ◽  
Author(s):  
Mike Janssen ◽  
Thomas Kamp ◽  
Jason Vander Woude
Keyword(s):  
Symbolic Power ◽  
Edge Ideal ◽  
Homogeneous Ideal ◽  
Edge Ideals ◽  
Ideal Formula ◽  
Number Of Generators ◽  
Recent Interest ◽  
Odd Cycle ◽  
Ordinary Power

Given a nontrivial homogeneous ideal [Formula: see text], a problem of great recent interest has been the comparison of the [Formula: see text]th ordinary power of [Formula: see text] and the [Formula: see text]th symbolic power [Formula: see text]. This comparison has been undertaken directly via an exploration of which exponents [Formula: see text] and [Formula: see text] guarantee the subset containment [Formula: see text] and asymptotically via a computation of the resurgence [Formula: see text], a number for which any [Formula: see text] guarantees [Formula: see text]. Recently, a third quantity, the symbolic defect, was introduced; as [Formula: see text], the symbolic defect is the minimal number of generators required to add to [Formula: see text] in order to get [Formula: see text]. We consider these various means of comparison when [Formula: see text] is the edge ideal of certain graphs by describing an ideal [Formula: see text] for which [Formula: see text]. When [Formula: see text] is the edge ideal of an odd cycle, our description of the structure of [Formula: see text] yields solutions to both the direct and asymptotic containment questions, as well as a partial computation of the sequence of symbolic defects.

scholarly journals Regular semigroups, fundamental semigroups and groups

1980 ◽  
Vol 29 (4) ◽  
pp. 475-503 ◽  
Author(s):  
D. B. McAlister
Keyword(s):  
Regular Semigroup ◽  
Partial Order ◽  
Direct Product ◽  
Homomorphic Image ◽  
Sufficient Conditions ◽  
Main Tool ◽  
Natural Partial Order ◽  
Regular Semigroups ◽  
Necessary And Sufficient ◽  
Fundamental Semigroups

AbstractIn this paper we obtain necessary and sufficient conditions on a regular semigroup in order that it should be an idempotent separating homomorphic image of a full subsemigroup of the direct product of a group and a fundamental or combinatorial regular semigroup. The main tool used is the concept of a prehomomrphism θ: S → T between regular semigroups. This is a mapping such that (ab) θ ≦ aθ bθ in the natural partial order on T.

On the homeotopy group of a 2-manifold (corrigendum)

1966 ◽  
Vol 62 (4) ◽  
pp. 679-681 ◽  
Author(s):  
W. B. R. Lickorish
Keyword(s):  
Infinite Number ◽  
Corrected Proof

A gap in the proof of the main theorem of (1) has been pointed out by C. D. Papakyriakopoulos and R. H. Fox. That proof was incomplete in that the table of homeomorphisms on page 776 of (1) did not cover all the possible cases (of which there are an infinite number). The corrected proof, given below, is intended to be a re-written version of the first half of section 3 of (1), the notation and references being those of (1).

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