Centralizing <i>n</i>‐Homoderivations of Semiprime Rings

El-Sayiad, M. S. Tammam; Ageeb, A.; Ghareeb, A.

doi:10.1155/2022/1112183

Cited by 3 publications

(1 citation statement)

References 14 publications

(13 reference statements)

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“…Here, I is a non-zero left ideal of R, and h is a homoderivation of R. In 2019, Al Harfien et al [5] and Rehman et al [6] have studied the commutativity of a semiprime (prime) ring admitting a homoderivation satisfying some identities on a ring. Researchers [7][8][9][10][11][12][13][14] have executed many noteworthy works concerning various properties of homoderivations during the last decades.…”

Section: H(xy) = H(x)h(y) + H(x)y + Xh(y)mentioning

confidence: 99%

Homoderivations in Prime Rings

ENGİN,

AYDIN

2023

Journal of New Theory

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The study consists of two parts. The first part shows that if $h_{1}(x)h_{2}(y)=h_{3}(x)h_{4}(y)$, for all $x,y\in R$, then $ h_{1}=h_{3}$ and $h_{2}=h_{4}$. Here, $h_{1},h_{2},h_{3},$ and $h_{4}$ are zero-power valued non-zero homoderivations of a prime ring $R$. Moreover, this study provide an explanation related to $h_{1}$ and $h_{2}$ satisfying the condition $ah_{1}+h_{2}b=0$. The second part shows that $L\subseteq Z$ if one of the following conditions is satisfied: $i. h(L)=(0)$, $ ii. h(L)\subseteq Z$, $iii. h(xy)=xy$, for all $x,y\in L$, $iv. h(xy)=yx$, for all $x,y\in L$, or $v. h([x,y])=0$, and for all $x,y\in L$. Here, $R$ is a prime ring with a characteristic other than $2$, $h$ is a homoderivation of $R$, and $L$ is a non-zero square closed Lie ideal of $R$.

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Section: H(xy) = H(x)h(y) + H(x)y + Xh(y)mentioning

confidence: 99%

Homoderivations in Prime Rings

ENGİN,

AYDIN

2023

Journal of New Theory

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On centrally-extended left derivations in rings

Gouda,

Nabiel

2024

Arab. J. Math.

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A Characterization of Semiprime Rings with Homoderivations

Söğütcü

2023

Journal of New Theory

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This paper is focused on the commutativity of the laws of semiprime rings, which satisfy some algebraic identities involving homoderivations on ideals. It provides new and notable results that will interest researchers in this field, such as “R contains a nonzero central ideal if R admits a nonzero homoderivation δ on I such that δ(I)⊆Z where R is a semiprime ring with center Z and I a nonzero ideal of R”. Moreover, the research also generalizes some results previously published in the literature, including derivation on prime rings using homoderivation semiprime rings. It also demonstrates the necessity of hypotheses operationalized in theorems by an example. Finally, the paper discusses how the results herein can be further developed in future research.

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Centralizing n‐Homoderivations of Semiprime Rings

Abstract: We introduce the notion of n -homoderivation on a ring ℜ and show that a semiprime ring ℜ must have a nontrivial central ideal if it admits an appropriate n -homoder… Show more

Cited by 3 publications

References 14 publications

Homoderivations in Prime Rings

Homoderivations in Prime Rings

On centrally-extended left derivations in rings

A Characterization of Semiprime Rings with Homoderivations

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