On $(δ,f)$-derivations and Jordan $(δ,f)$-derivations on modules
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909731464937472 |
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| author | Yanti, Gusti Ayu Dwi Wijayanti, Indah Emilia |
| author_facet | Yanti, Gusti Ayu Dwi Wijayanti, Indah Emilia |
| contents | Let $R$ be a ring with identity, $M,N$ right modules over $R$. An additive mapping $δ$ from $R$ to $R$ is called derivation on ring $R$ if it satisfies the Leibniz condition. If $δ$ is a derivation on $R$ and $f:M \rightarrow N$ is a module homomorphism over $R$, then an additive mapping $d:M \rightarrow N$ is called a $(δ,f)$-derivation if it satisfies $d(xa)=d(x)a+f(x)δ(a)$ for all $x \in M$ and $a \in R$. An additive mapping $δ: R \rightarrow R$ is called Jordan derivation on ring $R$ if $δ(x^2)=δ(x)x+xδ(x)$ for all $x \in R$, which is the generalization of derivation This paper presents generalization of Posner's First Theorem of $(δ,f)$-derivation on $2$-torsion prime modules. It also provides a generalization of some results in case of $2$-torsion free prime modules from ring situation. Moreover, we introduce a Jordan $(δ,f)$-derivation on modules and prove that every Jordan $(δ,f)$-derivation on modules is a $(δ,f)$-derivation on modules. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_07609 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On $(δ,f)$-derivations and Jordan $(δ,f)$-derivations on modules Yanti, Gusti Ayu Dwi Wijayanti, Indah Emilia Rings and Algebras 47B47, 13F20, 16S34 Let $R$ be a ring with identity, $M,N$ right modules over $R$. An additive mapping $δ$ from $R$ to $R$ is called derivation on ring $R$ if it satisfies the Leibniz condition. If $δ$ is a derivation on $R$ and $f:M \rightarrow N$ is a module homomorphism over $R$, then an additive mapping $d:M \rightarrow N$ is called a $(δ,f)$-derivation if it satisfies $d(xa)=d(x)a+f(x)δ(a)$ for all $x \in M$ and $a \in R$. An additive mapping $δ: R \rightarrow R$ is called Jordan derivation on ring $R$ if $δ(x^2)=δ(x)x+xδ(x)$ for all $x \in R$, which is the generalization of derivation This paper presents generalization of Posner's First Theorem of $(δ,f)$-derivation on $2$-torsion prime modules. It also provides a generalization of some results in case of $2$-torsion free prime modules from ring situation. Moreover, we introduce a Jordan $(δ,f)$-derivation on modules and prove that every Jordan $(δ,f)$-derivation on modules is a $(δ,f)$-derivation on modules. |
| title | On $(δ,f)$-derivations and Jordan $(δ,f)$-derivations on modules |
| topic | Rings and Algebras 47B47, 13F20, 16S34 |
| url | https://arxiv.org/abs/2508.07609 |