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arXiv · 2302.14571

Additivity of multiplicative (generalized) maps over rings

Sk Aziz·Arindam Ghosh·Om Prakash
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Abstract

In this paper, we mainly prove some results on the additivity of maps over rings under certain conditions. First, we discuss a special case of MARTINDALE III's theorem of \cite{1969M} as a bijective map $φ$ over a ring $R$ with a non-trivial idempotent satisfying $φ(ab)=φ(a)φ(b)$ for all $a, b\in R$, is additive. Then we prove that a map $D$ on $R$ satisfying $D(ab)=D(a)b+φ(a) D(b)$ for all $a,b\in R$, where $φ$ is the map mentioned above, is additive. Finally, we establish that if a map $g$ over $R$ satisfies $g(ab)=g(a)b+φ(a)D(b),$ for all $a,b\in R$ and the maps $φ$ and $D$ are mentioned above, then $g$ is additive.

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Sk Aziz, Arindam Ghosh, Om Prakash. 2025-10-04. Additivity of multiplicative (generalized) maps over rings. https://arxiv.org/abs/2302.14571

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