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Abstract
Let R be an associative ring with unit and let N(R) denote the set of nilpotent elements of R. R is said to be stronglyπ-regular if for each x∈R, there exist a positive integer n and an element y∈R such that x n=x n +1 y and xy=yx. R is said to be periodic if for each x∈R there are integers m,n≥ 1 such that m≠n and x m=x n. Assume that the idempotents in R are central. It is shown in this paper that R is a strongly π-regular ring if and only if N(R) coincides with the Jacobson radical of R and R/N(R) is regular. Some similar conditions for periodic rings are also obtained.
. Barman and C . Ray . Congruences for 𝑙-regular overpartitions and Andrews’ singular overpartitions . Ramanujan J . 45 : 497 – 515 , 2018 . [4] B. C . Berndt . Ramanujan’s Notebooks . Part III , Springer-Verlag , New York , 1991 . [5] W. Y
properties of the regular Coulomb wave function in [ 9 , 10 ], while the author investigated the zeros of regular Coulomb wave functions and their derivatives in [ 16 ]. Motivated by the above works our main aim is to investigate the lemniscate and
The total graph and regular graph of a commutative ring J. Pure Appl. Alg. 213 2224 – 2228 10.1016/j.jpaa.2009.03.013 . [2
Abstract
In this paper we study the congruences of *-regular semigroups, involution semigroups in which every element is p-related to a projection (an idempotent fixed by the involution). The class of *-regular semigroups was introduced by Drazin in 1979, as the involutorial counterpart of regular semigroups. In the standard approach to *-regular semigroup congruences, one ,starts with idempotents, i.e. with traces and kernels in the underlying regular semigroup, builds congruences of that semigroup, and filters those congruences which preserve the involution. Our approach, however, is more evenhanded with respect to the fundamental operations of *-regular semigroups. We show that idempotents can be replaced by projections when one passes from regular to *-regular semigroup congruences. Following the trace-kernel balanced view of Pastijn and Petrich, we prove that an appropriate equivalence on the set of projections (the *-trace) and the set of all elements equivalent to projections (the *-kernel) fully suffice to reconstruct an (involution-preserving) congruence of a *-regular semigroup. Also, we obtain some conclusions about the lattice of congruences of a *-regular semigroup.
Summary
We consider a generalisation of the Kurosh--Amitsur radical theory for rings (and more generally multi-operator groups) which applies to 0-regular varieties in which all operations preserve 0. We obtain results for subvarieties, quasivarieties and element-wise equationally defined classes. A number of examples of radical and semisimple classes in particular varieties are given, including hoops, loops and similar structures. In the first section, we introduce 0-normal varieties (0-regular varieties in which all operations preserve 0), and show that a key isomorphism theorem holds in a 0-normal variety if it is subtractive, a property more general than congruence permutability. We then define our notion of a radical class in the second section. A number of basic results and characterisations of radical and semisimple classes are then obtained, largely based on the more general categorical framework of L. M\'arki, R. Mlitz and R. Wiegandt as in [13]. We consider the subtractive case separately. In the third section, we obtain results concerning subvarieties and quasivarieties based on the results of the previous section, and also generalise to subtractive varieties some results for multi-operator group radicals defined by simple equational rules. Several examples of radical and semisimple classes are given for a range of fairly natural 0-normal varieties of algebras, most of which are subtractive.
Abstract
We define and study regular and locally closed sets in generalized topological spaces.
–the regular polygon case . Geombinatorics 31 : 49 – 67 , 2021 . [7] G . Exoo and D . Ismailescu . The chromatic number of the plane is at least 5: A new proof . Discrete & Computational Geometry , 1 – 11 , 2020 . [8] P . Frankl and R. M
Abstract
; and try to set some new criterion for repleteness and measure repleteness.A semigroup is called eventually regular if each of its elements has a regular power. In this paper we study certain fundamental congruences on an eventually regular semigroup. We generalize some results of Howie and Lallement (1966) and LaTorre (1983). In particular, we give a full description of the semilattice of group congruences (together with the least such a congruence) on an arbitrary eventually regular (orthodox) semigroup. Moreover, we investigate UBG-congruences on an eventually regular semigroup. Finally, we study the eventually regular subdirect products of an E-unitary semigroup and a Clifford semigroup.
