• Korean Journal of Mathematics
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• v.26 no.4
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• pp.799-808
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• 2018

In this paper, we extend the concept of double centralizer to triple centralizer and we show that, the triple centralizer is a $C^*$-algebra. Some algebraic properties are investigated.

• Journal of the Korean Mathematical Society
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• v.42 no.6
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• pp.1153-1167
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• 2005

In this paper, all rings and all near-rings R are associative, all modules are right R-modules. For a near-ring R, we consider representations of R as R-groups. We start with a study of AGR rings and their properties. Next, for any right R-module M, we define a new concept GM module and investigate the commutative property of faithful GM modules and some characterizations of GM modules. Similarly, for any near-ring R, we introduce an R-group with MR-property and some properties of MR groups.

• Bulletin of the Korean Mathematical Society
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• v.53 no.5
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• pp.1353-1362
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• 2016

Using the notion of complete nonabelian exterior square $G\hat{\wedge}G$ of a pro-p-group G (p prime), we develop the theory of the exterior degree $\hat{d}(G)$ in the infinite case, focusing on its relations with the probability of commuting pairs d(G). Among the main results of this paper, we describe upper and lower bounds for $\hat{d}(G)$ with respect to d(G). Here the size of the second homology group $H_2(G,\mathbb{Z}_p)$ (over the p-adic integers $\mathbb{Z}_p$) plays a fundamental role. A further result of homological nature is placed at the end, in order to emphasize the influence of $H_2(G,\mathbb{Z}_p)$ both on G and $\hat{d}(G)$.

• Journal of Applied and Pure Mathematics
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• v.4 no.3_4
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• pp.99-105
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• 2022

The objective of this research paper is to prove that an additive mapping T from a semiprime ring R to itself will be centralizer having a suitable torsion restriction on R if it satisfy any one of the following algebraic equations (a) 2T(xⁿyⁿxⁿ) = T(xⁿ)yⁿxⁿ + xⁿyⁿT(xⁿ) (b) 3T(xⁿyⁿxⁿ) = T(xⁿ)yⁿxⁿ+xⁿT(yⁿ)xⁿ+xⁿyⁿT(xⁿ) for every x, y ∈ R. Further, few extensions of these results are also presented in the framework of *-ring.

• Journal of applied mathematics & informatics
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• v.40 no.3_4
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• pp.709-717
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• 2022

The objective of this research is to prove that an additive mapping T : R → R is a left as well as right centralizer on R if it satisfies any one of the following identities: (i) T(xⁿyⁿ + yⁿxⁿ) = T(xⁿ)yⁿ + yⁿT(xⁿ) (ii) 2T(xⁿyⁿ) = T(xⁿ)yⁿ + yⁿT(xⁿ) for each x, y ∈ R, where n ≥ 1 is a fixed integer and R is any n!-torsion free semiprime ring. In addition, we talk over above identities in the setting of *-ring(ring with involution).

• Journal of the Korean Mathematical Society
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• v.54 no.2
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• pp.685-696
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• 2017

A linear mapping ${\phi}$ on an algebra $\mathcal{A}$ is called a centralizable mapping at $G{\in}{\mathcal{A}}$ if ${\phi}(AB)={\phi}(A)B= A{\phi}(B)$ for each A and B in $\mathcal{A}$ with AB = G, and ${\phi}$ is called a derivable mapping at $G{\in}{\mathcal{A}}$ if ${\phi}(AB)={\phi}(A)B+A{\phi}(B)$ for each A and B in $\mathcal{A}$ with AB = G. A point G in A is called a full-centralizable point (resp. full-derivable point) if every centralizable (resp. derivable) mapping at G is a centralizer (resp. derivation). We prove that every point in a von Neumann algebra or a triangular algebra is a full-centralizable point. We also prove that a point in a von Neumann algebra is a full-derivable point if and only if its central carrier is the unit.