ALGEBRAIC CONDITIONS WITH HOMODERIVATIONS UNDER WHICH NEAR RINGS ARE RINGS

Document Type : Original Manuscript

Authors

1 Department of Mathematics, Chandigarh University, Mohali, Punjab, India.

2 Department of Mathematics, Aligarh Muslim University, Aligarh, India.

10.22044/jas.2025.14519.1837

Abstract

An additive mapping $\Gamma:\mathfrak{N}\rightarrow \mathfrak{N}$ is called a homoderivation on a near ring $\mathfrak{N}$ if $\Gamma(uv)=\Gamma(u)\Gamma(v)+\Gamma(u)v+u\Gamma(v)$ for all $u,v\in \mathfrak{N}$. The aim of the present paper is to explore the following algebraic identities: (i) $\Gamma([j,u])=[j,u]$ for all $j\in \mathfrak{J},\;u\in \mathfrak{U}$, (ii) $\Gamma([j,u])+[j,u]=(j\circ u)$ for all $j\in \mathfrak{J},\;u\in \mathfrak{U}$, (iii) $\Gamma(j\circ u)+(j\circ u)=[j,u]$ for all $j\in \mathfrak{J},\;u\in \mathfrak{U}$, (iv) $\Gamma([j,n])= 0$ for all $j\in \mathfrak{J},\;n\in \mathfrak{N}$, (v) $\Gamma(jn)+jn=[j,n]$ for all $j\in \mathfrak{J},\;n\in \mathfrak{N}$, (vi) $\Gamma(jn)+jn=(j\circ n)$ for all $j\in \mathfrak{J},\;n\in \mathfrak{N}$, where $\mathfrak{J}$ is a nonzero Jordan ideal of $\mathfrak{N}$ and $\mathfrak{U}$, a nonzero semigroup ideal of $\mathfrak{N}$. In addition, we present examples to support the hypotheses of our
results.

Keywords

References

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Journal of Algebraic Systems
Volume 14, Issue 4
September 2026
Pages 603-612

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