• Korean Journal of Mathematics
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• v.29 no.3
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• pp.483-492
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• 2021

This article concerns the property that for any element a in a ring, if a²ⁿ = aⁿ for some n ≥ 2 then a² = a. The class of rings with this property is large, but there also exist many kinds of rings without that, for example, rings of characteristic ≠2 and finite fields of characteristic ≥ 3. Rings with such a property is called reduced-over-idempotent. The study of reduced-over-idempotent rings is based on the fact that the characteristic is 2 and every nonzero non-identity element generates an infinite multiplicative semigroup without identity. It is proved that the reduced-over-idempotent property pass to polynomial rings, and we provide power series rings with a partial affirmative argument. It is also proved that every finitely generated subring of a locally finite reduced-over-idempotent ring is isomorphic to a finite direct product of copies of the prime field {0, 1}. A method to construct reduced-over-idempotent fields is also provided.

• Bulletin of the Korean Mathematical Society
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• v.60 no.6
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• pp.1463-1475
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• 2023

In this paper, we introduce a class of rings generalizing unit regular rings and being a subclass of semipotent rings, which is called idempotent unit regular. We call a ring R an idempotent unit regular ring if for all r ∈ R - J(R), there exist a non-zero idempotent e and a unit element u in R such that er = eu, where this condition is left and right symmetric. Thus, we have also that there exist a non-zero idempotent e and a unit u such that ere = eue for all r ∈ R - J(R). Various basic characterizations and properties of this class of rings are proved and it is given the relationships between this class of rings and some well-known classes of rings such as semiperfect, clean, exchange and semipotent. Moreover, we obtain some results about when the endomorphism ring of a module in a class of left R-modules X is idempotent unit regular.

• Kyungpook Mathematical Journal
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• v.54 no.3
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• pp.501-509
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• 2014

In 2000, K. Denecke and K. Mahdavi showed that there are many idempotent elements in $Hyp_{N_{\varphi}}(V)$ the set of normal form hypersubstitutions of type ${\tau}=(2)$ which are not idempotent elements in Hyp(2) the set of all hypersubstitutions of type ${\tau}=(2)$. They considered in which varieties, idempotent elements of Hyp(2) are idempotent elements of $Hyp_{N_{\varphi}}(V)$. In this paper, we study the similar problems on the set of all generalized hypersubstitutions of type ${\tau}=(2)$ and the set of all normal form generalize hypersubstitutions of type ${\tau}=(2)$ and determine the order of normal form generalize hypersubstitutions of type ${\tau}=(2)$.

• Communications of the Korean Mathematical Society
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• v.12 no.1
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• pp.45-50
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• 1997

The purpose of this paper is to give a characterization of automorphisms of the weighted Lotka-Volterra algebra $(A,\omega)$ at idempotent elements and to offer a condition that $(A,\omege)$ becomes a Jordan algebra.

• Communications of the Korean Mathematical Society
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• v.33 no.3
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• pp.751-757
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• 2018

A ring is nil-clean if every element is a sum of a nilpotent and an idempotent, and a ring is involution-clean if every element is a sum of an involution and an idempotent. In this paper, a description of nil-clean rings of nilpotency index at most 2 is obtained, and is applied to improve a known result on involution-clean rings.

• Kyungpook Mathematical Journal
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• v.59 no.1
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• pp.35-45
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• 2019

The ${\Gamma}$-semihyperring is a generalization of the concepts of a semiring, a semihyperring and a ${\Gamma}$-semiring. Here, the notions of (strongly) regular ${\Gamma}$-semihyperring, idempotent ${\Gamma}$-semihyperring; invertible set, invertible element in a ${\Gamma}$-semihyperring are introduced, and several examples given. It is proved that if all subsets of ${\Gamma}$-semihyperring are strongly regular then for every ${\Delta}{\subseteq}{\Gamma}$, there is a ${\Delta}$-idempotent subset of R. Regularity conditions of ${\Gamma}$-semihyperrings in terms of ideals of ${\Gamma}$-semihyperrings are also characterized.

• Kyungpook Mathematical Journal
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• v.49 no.3
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• pp.457-471
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• 2009

Let S = {a, b, c, ...} and ${\Gamma}$ = {${\alpha}$, ${\beta}$, ${\gamma}$, ...} be two nonempty sets. S is called a ${\Gamma}$-semigroup if $a{\alpha}b{\in}S$, for all ${\alpha}{\in}{\Gamma}$ and a, b ${\in}$ S and $(a{\alpha}b){\beta}c=a{\alpha}(b{\beta}c)$, for all a, b, c ${\in}$ S and for all ${\alpha}$, ${\beta}$ ${\in}$ ${\Gamma}$. An element $e{\in}S$ is said to be an ${\alpha}$-idempotent for some ${\alpha}{\in}{\Gamma}$ if $e{\alpha}e$ = e. A ${\Gamma}$-semigroup S is called an E-inversive ${\Gamma}$-semigroup if for each $a{\in}S$ there exist $x{\in}S$ and ${\alpha}{\in}{\Gamma}$ such that a${\alpha}$x is a ${\beta}$-idempotent for some ${\beta}{\in}{\Gamma}$. A ${\Gamma}$-semigroup is called a right E-${\Gamma}$-semigroup if for each ${\alpha}$-idempotent e and ${\beta}$-idempotent f, $e{\alpha}$ is a ${\beta}$-idempotent. In this paper we investigate different properties of E-inversive ${\Gamma}$-semigroup and right E-${\Gamma}$-semigroup.

• Journal of the Korean Mathematical Society
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• v.57 no.3
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• pp.793-807
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• 2020

Stable range of rings is a unifying concept for problems related to the substitution and cancellation of modules. The newly appeared element-wise setting for the simplest case of stable range one is tempting to study the lifting property modulo ideals. We study the lifting of elements having (idempotent) stable range one from a quotient of a ring R modulo a two-sided ideal I by providing several examples and investigating the relations with other lifting properties, including lifting idempotents, lifting units, and lifting of von Neumann regular elements. In the case where the ring R is a left or a right duo ring, we show that stable range one elements lift modulo every two-sided ideal if and only if R is a ring with stable range one. Under a mild assumption, we further prove that the lifting of elements having idempotent stable range one implies the lifting of von Neumann regular elements.

• Journal of Applied and Pure Mathematics
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• v.5 no.1_2
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• pp.1-8
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• 2023

This study is on a Boolean B or Boolean lattice L in abstract algebra with closed binary operation *, complement and distributive properties. Both Binary operations and logic properties dominate this set. A lattice sheds light on binary operations and other algebraic structures. In particular, the construction of the elements of this L set from idempotent elements, our definition of k-order idempotent has led to the expanded definition of the definition of the lattice theory. In addition, a lattice offers clever solutions to vital problems in life with the concept of logic. The restriction on a lattice is clearly also limit such applications. The flexibility of logical theories adds even more vitality to practices. This is the main theme of the study. Therefore, the properties of the set elements resulting from the binary operation force the logic theory. According to the new definition given, some properties, lemmas and theorems of the lattice theory are examined. Examples of different situations are given.

• Bulletin of the Korean Mathematical Society
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• v.48 no.4
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• pp.759-767
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• 2011

An element of a ring is called strongly nil clean provided that it can be written as the sum of an idempotent and a nilpotent element that commute. A ring is strongly nil clean in case each of its elements is strongly nil clean. We investigate, in this article, the strongly nil cleanness of 2${\times}$2 matrices over local rings. For commutative local rings, we characterize strongly nil cleanness in terms of solvability of quadratic equations. The strongly nil cleanness of a single triangular matrix is studied as well.