• East Asian mathematical journal
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• v.18 no.2
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• pp.271-282
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• 2002

Let R be a ring with an endomorphism $\sigma$ and a derivation $\delta$. An ideal I of R is ($\sigma,\;\delta$)-ideal of R if $\sigma(I){\subseteq}I$ and $\delta(I){\subseteq}I$. An ideal P of R is a ($\sigma,\;\delta$)-prime ideal of R if P(${\neq}R$) is a ($\sigma,\;\delta$)-ideal and for ($\sigma,\;\delta$)-ideals I and J of R, $IJ{\subseteq}P$ implies that $I{\subseteq}P$ or $J{\subseteq}P$. An ideal Q of R is ($\sigma,\;\delta$)-semiprime ideal of R if Q is a ($\sigma,\;\delta$)-ideal and for ($\sigma,\;\delta$)-ideal I of R, $I^2{\subseteq}Q$ implies that $I{\subseteq}Q$. The ($\sigma,\;\delta$)-prime radical (resp. prime radical) is defined by the intersection of all ($\sigma,\;\delta$)-prime ideals (resp. prime ideals) of R and is denoted by $P_{(\sigma,\delta)}(R)$(resp. P(R)). In this paper, the following results are obtained: (1) $P_{(\sigma,\delta)}(R)$ is the smallest ($\sigma,\;\delta$)-semiprime ideal of R; (2) For every extended endomorphism $\bar{\sigma}$ of $\sigma$, the $\bar{\sigma}$-prime radical of an Ore extension $P(R[x;\sigma,\delta])$ is equal to $P_{\sigma,\delta}(R)[x;\sigma,\delta]$.

• Honam Mathematical Journal
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• v.23 no.1
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• pp.1-4
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• 2001

In [2, 7.3.2], the set of attached prime ideals of local cohomology module $H_m^n(M)$ were calculated, where (A, m) be Noetherian local ring, M finite A-module and $dim_A(M)=n$, and also in the special case in which furthermore A is a homomorphic image of a Gornestien local ring (A', m') (see [2, 11.3.6]). In this paper, we shall obtain this set, by another way in this special case.

• Bulletin of the Korean Mathematical Society
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• v.48 no.5
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• pp.917-922
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• 2011

Let R be a prime ring, H a generalized derivation of R, L a noncentral Lie ideal of R, and 0 ${\neq}$ a ${\in}$ R. Suppose that $au^sH(u)u^t$ = 0 for all u ${\in}$ L, where s; t ${\geq}$ 0 are fixed integers. Then H = 0 unless satisfies $S_4$, the standard identity in four variables.

• Bulletin of the Korean Mathematical Society
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• v.46 no.3
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• pp.599-605
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• 2009

Let R be a prime ring, H a generalized derivation of R and L a noncommutative Lie ideal of R. Suppose that $u^sH(u)u^t$ = 0 for all u $\in$ L, where s $\geq$ 0, t $\geq$ 0 are fixed integers. Then H(x) = 0 for all x $\in$ R unless char R = 2 and R satisfies $S_4$, the standard identity in four variables.

• Kyungpook Mathematical Journal
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• v.18 no.1
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• pp.105-107
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• 1978

Considering the prime z-filters on a topological space X through the structures of the ring C(X) of continuous functions. a prime z-filter is uniquely determined by a primary z-ideal in the ring C(X), i. e., they have a one-to-one correspondence. Any primary ideal is contained in a unique maximal ideal in C(X). Denoting $\mathfrak{F}(X)$, $\mathfrak{Q}(X)$, 𝔐(X) the prime, primary-z, maximal spectra, respectively, $\mathfrak{Q}(X)$ is neither an open nor a closed subspace of $\mathfrak{F}(X)$.

• Communications of the Korean Mathematical Society
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• v.36 no.4
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• pp.641-650
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• 2021

The notions of skew-commuting/commuting/semi-commuting/skew-centralizing/semi-centralizing mappings play an important role in ring theory. ${\mathfrak{C}}^*$-algebras with these properties have been studied considerably less and the existing results are motivating the researchers. This article elaborates the structure of prime rings and ${\mathfrak{C}}^*$-algebras satisfying certain functional identities involving automorphisms.

• Journal of applied mathematics & informatics
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• v.39 no.5_6
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• pp.761-768
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• 2021

Let I be a nonzero ideal of a prime ring R and m, n are positive integers. If R admits a generalized derivation F satisfying F(x)ⁿF(y)^m - xⁿy^m = 0 for any y, x ∈ I, then F(x) = ax for any x ∈ R and a^m+n = 1, where a ∈ C, the extended centroid of R.

• Kyungpook Mathematical Journal
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• v.61 no.4
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• pp.727-735
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• 2021

In this paper, we will study the structure of the quotient ring R/P of an arbitrary ring R by a prime ideal P. We do so using differential identities involving generalized derivations of R. We enrich our results with examples that show the necessity of their assumptions.

• Journal of Applied and Pure Mathematics
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• v.6 no.1_2
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• pp.13-20
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• 2024

Let R be a ring with involution. In the present paper, we characterize biadditive mappings which satisfies some functional identities related to symmetric Jordan (𝛼, 1)^*-biderivation of prime rings with involution. In particular, we prove that on a 2-torsion free prime ring with involution, every symmetric Jordan triple (𝛼, 1)^*-biderivation is a symmetric Jordan (𝛼, 1)^*-biderivation.

• Honam Mathematical Journal
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• v.31 no.2
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• pp.203-217
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• 2009

In this note, we introduce a symmetric generalized 3-derivation in near-rings and investigate some conditions for a nearring to be a commutative ring.