Additivity of multiplicative (generalized) maps over rings
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866916988439232512 |
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| author | Aziz, Sk Ghosh, Arindam Prakash, Om |
| author_facet | Aziz, Sk Ghosh, Arindam Prakash, Om |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_14571 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Additivity of multiplicative (generalized) maps over rings Aziz, Sk Ghosh, Arindam Prakash, Om Rings and Algebras 16W10, 16W25, 16W55, 17C27 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. |
| title | Additivity of multiplicative (generalized) maps over rings |
| topic | Rings and Algebras 16W10, 16W25, 16W55, 17C27 |
| url | https://arxiv.org/abs/2302.14571 |