• Korean Journal of Mathematics
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• v.21 no.4
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• pp.463-471
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• 2013

Let R be a ring with identity 1, $I(R){\neq}\{0\}$ be the set of all nonunit idempotents in R, and M(R) be the set of all primitive idempotents and 0 of R. We say that I(R) is additive if for all e, $f{\in}I(R)$ ($e{\neq}f$), $e+f{\in}I(R)$. In this paper, the following are shown: (1) I(R) is a finite additive set if and only if $M(R){\backslash}\{0\}$ is a complete set of primitive central idempotents, char(R) = 2 and every nonzero idempotent of R can be expressed as a sum of orthogonal primitive idempotents of R; (2) for a regular ring R such that I(R) is a finite additive set, if the multiplicative group of all units of R is abelian (resp. cyclic), then R is a commutative ring (resp. R is a finite direct product of finite field).

• Journal of applied mathematics & informatics
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• v.18 no.1_2
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• pp.25-43
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• 2005

Let F be a finite field of prime power order q(odd) and the multiplicative order of q modulo $2^{n}\;(n>1)\;be\; {\phi}(2^{n})/2$. If n > 3, then q is odd number(prime or prime power) of the form $8m{\pm}3$. If q = 8m - 3, then the ring $R_{2^n} = F[x]/ < x^{2^n}-1 >$ has 2n primitive idempotents. The explicit expressions for these primitive idempotents are obtained and the minimal QR cyclic codes of length $2^{n}$ generated by these idempotents are completely described. If q = 8m + 3 then the expressions for the 2n - 1 primitive idempotents of $R_{2^n}$ are obtained. The generating polynomials and the upper bounds of the minimum distance of minimal QR cyclic codes generated by these 2n-1 idempotents are also obtained. The case n = 2,3 is dealt separately.

• Kyungpook Mathematical Journal
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• v.45 no.2
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• pp.171-175
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• 2005

In this paper, we introduce the concept of op-idempotents. It is shown that every exchange ring can be characterized by op-idempotents

• Bulletin of the Korean Mathematical Society
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• v.50 no.6
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• pp.1957-1972
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• 2013

The reflexive property for ideals was introduced by Mason and has important roles in noncommutative ring theory. In this note we study the structure of idempotents satisfying the reflexive property and introduce reflexive-idempotents-property (simply, RIP) as a generalization. It is proved that the RIP can go up to polynomial rings, power series rings, and Dorroh extensions. The structure of non-Abelian RIP rings of minimal order (with or without identity) is completely investigated.

• Communications of the Korean Mathematical Society
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• v.13 no.4
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• pp.751-757
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• 1998

Using idempotents in quasi-lattices, we show that the category Latt of lattices is both reflective and coreflective in the category qLatt of quasi-lattices and homomorphisms. It is also shown that a quasi-ordered set is a quasi-lattice iff its partially ordered reflection is a lattice.

• Bulletin of the Korean Mathematical Society
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• v.48 no.5
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• pp.1033-1039
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• 2011

Let R be a ring with identity 1, I(R) be the set of all idempotents in R and G be the group of all units of R. In this paper, we show that for any semiperfect ring R in which 2 = 1+1 is a unit, I(R) is closed under multiplication if and only if R is a direct sum of local rings if and only if the set of all minimal idempotents in R is closed under multiplication and eGe is contained in the group of units of eRe. In particular, for a left Artinian ring in which 2 is a unit, R is a direct sum of local rings if and only if the set of all minimal idempotents in R is closed under multiplication.

• Journal of the Korean Mathematical Society
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• v.51 no.3
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• pp.463-472
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• 2014

Let R be a ring with identity 1, I(R) be the set of all nonunit idempotents in R and $S_{\ell}$(R) (resp. $S_r$(R)) be the set of all left (resp. right) semicentral idempotents in R. In this paper, the following are investigated: (1) $e{\in}S_{\ell}(R)$ (resp. $e{\in}S_r(R)$) if and only if re=ere (resp. er=ere) for all nilpotent elements $r{\in}R$ if and only if $fe{\in}I(R)$ (resp. $ef{\in}I(R)$) for all $f{\in}I(R)$ if and only if fe=efe (resp. ef=efe) for all $f{\in}I(R)$ if and only if fe=efe (resp. ef=efe) for all $f{\in}I(R)$ which are isomorphic to e if and only if $(fe)^n=(efe)^n$ (resp. $(ef)^n=(efe)^n$) for all $f{\in}I(R)$ which are isomorphic to e where n is some positive integer; (2) For a ring R having a complete set of centrally primitive idempotents, every nonzero left (resp. right) semicentral idempotent is a finite sum of orthogonal left (resp. right) semicentral primitive idempotents, and eRe has also a complete set of primitive idempotents for any $0{\neq}e{\in}S_{\ell}(R)$ (resp. 0$0{\neq}e{\in}S_r(R)$).

• Communications of the Korean Mathematical Society
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• v.10 no.1
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• pp.123-131
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• 1995

The notion of our non-associative algebra is obtained from the Lotka-Volterra system of differential equation describing competitiion between animals or vegetals species and also in the kinetic theory of gasses. For the structure of an algebra, the existence of idempotents is of particular interest. But also from the biological aspect the existence of such elements is of interest because the equilibria of a population which can be described by an algebra correspond to idempotents of this algebra. Thus we present some properties of the general natures for a Lotka-Volterra algebra associated to a weight function and idempotents elements.

• Honam Mathematical Journal
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• v.41 no.2
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• pp.321-328
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• 2019

We completely characterize the isomorphic class of those associative unitary rings whose elements are sums of four commuting idempotents. Our main theorem enlarges results due to Hirano-Tominaga (Bull. Austral. Math. Soc., 1988), Tang et al. (Lin. & Multilin. Algebra, 2019), Ying et al. (Can. Math. Bull., 2016) as well as results due to the author in (Alban. J. Math., 2018), (Gulf J. Math., 2018), (Bull. Iran. Math. Soc., 2018) and (Boll. Un. Mat. Ital., 2019).

• Bulletin of the Korean Mathematical Society
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• v.60 no.5
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• pp.1321-1334
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• 2023

An idempotent e of a ring R is called right (resp., left) semicentral if er = ere (resp., re = ere) for any r ∈ R, and an idempotent e of R∖{0, 1} will be called right (resp., left) quasicentral provided that for any r ∈ R, there exists an idempotent f = f(e, r) ∈ R∖{0, 1} such that er = erf (resp., re = fre). We show the whole shapes of idempotents and right (left) semicentral idempotents of upper triangular matrix rings and polynomial rings. We next prove that every nontrivial idempotent of the n by n full matrix ring over a principal ideal domain is right and left quasicentral and, applying this result, we can find many right (left) quasicentral idempotents but not right (left) semicentral.