• Journal of the Korean Mathematical Society
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• v.48 no.5
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• pp.927-938
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• 2011

Let G be an abelian monoid with identity e. Let R be a G-graded commutative ring, and M a graded R-module. In this paper we first introduce the concept of graded primal submodules of M an give some basic results concerning this class of submodules. Then we characterize the graded primal ideals of the idealization R(+)M.

• The Pure and Applied Mathematics
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• v.2 no.1
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• pp.31-34
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• 1995

Let R$^n$be n-th Euclidean space. Let be the n-th spere embeded as a subspace in R$\^$n+1/ centered at the origin. In this paper, we are going to consider the function space F = {f│f : S$^n$\longrightarrow S$^n$} metrized by as follow D(f,g)=d(f($\chi$), g($\chi$)) where f, g $\in$ F and d is the metric in S$^n$. Finally we want to find certain relation these spaces.(omitted)

• Kyungpook Mathematical Journal
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• v.46 no.2
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• pp.169-180
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• 2006

This paper proves the following results: Let $f$ be a transcendental entire function, and let $k({\geq})2$ be a positive integer. If $T(r,\;f){\neq}N_{1)}(r,1/f)+S(r,\;f)$, then $ff^{(k)}$ assumes every finite nonzero value infinitely often. Also the case when f is a transcendental meromorphic function has been considered and some results are obtained.

• Journal of applied mathematics & informatics
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• v.39 no.3_4
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• pp.259-266
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• 2021

Let R be a commutative ring with identity and let S be a nonempty subset of R. We define S to be an almost multiplicative subset of R if for each a, b ∈ S, there exist integers m, n ≥ 1 such that a^mbⁿ ∈ S. In this article, we study some utilization of almost multiplicative subsets.

• Journal of Applied and Pure Mathematics
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• v.6 no.3_4
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• pp.155-165
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• 2024

Let R be a semi-prime ring. Let [δ₁, D₁] and [δ₂, D₂] be two generalized symmetric reverse biderivations of R with associated reverse biderivations D₁ and D₂. The main aim of the present paper is to establish conditions of orthogonality for symmetric reverse biderivations and symmetric generalized reverse biderivations in R.

• Journal of the Korean Mathematical Society
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• v.53 no.2
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• pp.403-414
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• 2016

Let R be a ring, and let M be a left R-module. If M is Rad-supplementing, then every direct summand of M is Rad-supplementing, but not each factor module of M. Any finite direct sum of Rad-supplementing modules is Rad-supplementing. Every module with composition series is (Rad-)supplementing. M has a Rad-supplement in its injective envelope if and only if M has a Rad-supplement in every essential extension. R is left perfect if and only if R is semilocal, reduced and the free left R-module $(_RR)^{({\mathbb{N})}$ is Rad-supplementing if and only if R is reduced and the free left R-module $(_RR)^{({\mathbb{N})}$ is ample Rad-supplementing. M is ample Rad-supplementing if and only if every submodule of M is Rad-supplementing. Every left R-module is (ample) Rad-supplementing if and only if R/P(R) is left perfect, where P(R) is the sum of all left ideals I of R such that Rad I = I.

• Bulletin of the Korean Mathematical Society
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• v.49 no.4
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• pp.885-898
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• 2012

Let m, n, r be nonzero fixed positive integers, R a 2-torsion free prime ring, Q its right Martindale quotient ring, and L a non-central Lie ideal of R. Let D : $R{\rightarrow}R$ be a skew derivation of R and $E(x)=D(x^{m+n+r})-D(x^m)x^{n+r}-x^mD(x^n)x^r-x^{m+n}D(x^r)$. We prove that if $E(x)=0$ for all $x{\in}L$, then D is a usual derivation of R or R satisfies $s_4(x_1,{\ldots},x_4)$, the standard identity of degree 4.

• Kyungpook Mathematical Journal
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• v.62 no.3
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• pp.595-613
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• 2022

Let R be a commutative ring with identity and let I be a proper ideal of R. In this paper, we study the ideal based extended zero-divisor graph 𝚪'_I (R) and prove that 𝚪'_I (R) is connected with diameter at most two and if 𝚪'_I (R) contains a cycle, then girth is at most four girth at most four. Furthermore, we study affinity the connection between the ideal based extended zero-divisor graph 𝚪'_I (R) and the ideal-based zero-divisor graph 𝚪_I (R) associated with the ideal I of R. Among the other things, for a radical ideal of a ring R, we show that the ideal-based extended zero-divisor graph 𝚪'_I (R) is identical to the ideal-based zero-divisor graph 𝚪_I (R) if and only if R has exactly two minimal prime-ideals which contain I.

• Kyungpook Mathematical Journal
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• v.64 no.3
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• pp.417-421
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• 2024

Let R be a commutative ring and let M be an R-module. In this note, we give a brief proof of the Hilbert basis theorem for Noetherian modules. This states that if R contains the identity and M is a Noetherian unitary R-module, then M[X] is a Noetherian R[X]-module. We also show that if M[X] is a Noetherian R[X]-module, then M is a Noetherian R-module and there exists an element e ∈ R such that em = m for all m ∈ M. Finally, we prove that if M[X] is a Noetherian R[X]-module and ann_R(M) = (0), then R has the identity and M is a unitary R-module.

• Journal of the Korean Mathematical Society
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• v.32 no.3
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• pp.519-529
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• 1995

Let M be a compact smooth manifold of dimension n and let E be a flat (complet) vector bundle over M of rank r.