• Kyungpook Mathematical Journal
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• v.55 no.1
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• pp.21-28
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• 2015

Let R be a prime ring with center Z, I a nonzero ideal of R, and ${\sigma}$ a non-trivial automorphism of R such that $\{(x{\circ}y)^{\sigma}-(x{\circ}y)\}^n{\in}Z$ for all $x,y{\in}I$. If either char(R) > n or char (R) = 0, then R satisfies $s_4$, the standard identity in 4 variables.

• Communications of the Korean Mathematical Society
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• v.32 no.4
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• pp.827-833
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• 2017

In this paper, we define a set including of all $f_a$ with $a{\in}R$ generalized derivations of R and is denoted by $f_R$. It is proved that (i) the mapping $g:L(R){\rightarrow}f_R$ given by g (a) = f-a for all $a{\in}R$ is a Lie epimorphism with kernel $N_{{\sigma},{\tau}}$ ; (ii) if R is a semiprime ring and ${\sigma}$ is an epimorphism of R, the mapping $h:f_R{\rightarrow}I(R)$ given by $h(f_a)=i_{{\sigma}(-a)}$ is a Lie epimorphism with kernel $l(f_R)$ ; (iii) if $f_R$ is a prime Lie ring and A, B are Lie ideals of R, then $[f_A,f_B]=(0)$ implies that either $f_A=(0)$ or $f_B=(0)$.

• Kyungpook Mathematical Journal
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• v.55 no.1
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• pp.41-50
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• 2015

The purpose of this paper is to prove some results which are of independent interest and related to additive maps on ${\sigma}$-prime rings. Further, examples are given to demonstrate that the restrictions imposed on the hypotheses of these results are not superfluous.

• Communications of the Korean Mathematical Society
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• v.27 no.3
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• pp.441-448
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• 2012

Let R be a prime ring with center Z and characteristic different from two, U a nonzero Lie ideal of R such that $u^2{\in}U$ for all $u{\in}U$ and $d$ be a nonzero (${\sigma}$, ${\tau}$)-derivation of R. We prove the following results: (i) If $[d(u),u]_{{\sigma},{\tau}}$ = 0 or $[d(u),u]_{{\sigma},{\tau}}{\in}C_{{\sigma},{\tau}}$ for all $u{\in}U$, then $U{\subseteq}Z$. (ii) If $a{\in}R$ and $[d(u),a]_{{\sigma},{\tau}}$ = 0 for all $u{\in}U$, then $U{\subseteq}Z$ or $a{\in}Z$. (iii) If $d([u,v])={\pm}[u,v]_{{\sigma},{\tau}}$ for all $u{\in}U$, then $U{\subseteq}Z$.

• Bulletin of the Korean Mathematical Society
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• v.50 no.6
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• pp.1863-1871
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• 2013

For a ring R with an automorphism ${\sigma}$, an n-additive mapping ${\Delta}:R{\times}R{\times}{\cdots}{\times}R{\rightarrow}R$ is called a skew n-derivation with respect to ${\sigma}$ if it is always a ${\sigma}$-derivation of R for each argument. Namely, if n - 1 of the arguments are fixed, then ${\Delta}$ is a ${\sigma}$-derivation on the remaining argument. In this short note, from Bre$\check{s}$ar Theorems, we prove that a skew n-derivation ($n{\geq}3$) on a semiprime ring R must map into the center of R.

• Journal of the Korean Mathematical Society
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• v.50 no.6
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• pp.1271-1290
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• 2013

Let M be a fixed left R-module. For a left R-module X, we introduce the notion of M-prime (resp. M-semiprime) submodule of X such that in the case M=R, it coincides with prime (resp. semiprime) submodule of X. Other concepts encountered in the general theory are M-$m$-system sets, M-$n$-system sets, M-prime radical and M-Baer's lower nilradical of modules. Relationships between these concepts and basic properties are established. In particular, we identify certain submodules of M, called "primeM-ideals", that play a role analogous to that of prime (two-sided) ideals in the ring R. Using this definition, we show that if M satisfies condition H (defined later) and $Hom_R(M,X){\neq}0$ for all modules X in the category ${\sigma}[M]$, then there is a one-to-one correspondence between isomorphism classes of indecomposable M-injective modules in ${\sigma}[M]$ and prime M-ideals of M. Also, we investigate the prime M-ideals, M-prime submodules and M-prime radical of Artinian modules.

• Communications of the Korean Mathematical Society
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• v.38 no.3
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• pp.679-693
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• 2023

Let 𝓡 be a 𝜎-prime ring with involution 𝜎. The main objective of this paper is to describe the structure of the 𝜎-prime ring 𝓡 with involution 𝜎 satisfying certain differential identities involving three derivations 𝜓₁, 𝜓₂ and 𝜓₃ such that 𝜓₁[t₁, 𝜎(t₁)] + [𝜓₂(t₁), 𝜓₂(𝜎(t₁))] + [𝜓₃(t₁), 𝜎(t₁)] ∈ 𝒥_Z for all t₁ ∈ 𝓡. Further, some other related results have also been discussed.

• Communications of the Korean Mathematical Society
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• v.22 no.4
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• pp.487-491
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• 2007

In this paper, we introduce a symmetric $bi-({\sigma},\;{\tau})-derivation$ in a near-ring and generalize some of the results in [5, 6, 8, 9].

• Kyungpook Mathematical Journal
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• v.50 no.3
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• pp.379-387
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• 2010

In this paper, we present some results concerning orthogonal generalized (${\sigma},{\tau}$)-derivations in semiprime near-rings. These results are a generalization of result of Bresar and Vukman, which are related to a theorem of Posner for the product of two derivations in prime rings.