Identities on additive mappings in semiprime rings

A. Z.  Ansari; N. Rehman

doi:10.30970/ms.58.2.133-141

A. Z. Ansari Department of Mathematics Faculty of Science Islamic University in Madinah, K.S.A Madinah, India
N. Rehman Department of Mathematics, Faculty of Science Aligarh Muslim University, Aligarh, India

DOI: https://doi.org/10.30970/ms.58.2.133-141

Keywords: semiprime rings; generalized derivation; generalized left derivation and additive mappings

Abstract

Consider a ring $R$, which is semiprime and also having $k$-torsion freeness. If $F, d : R\to R$ are two additive maps fulfilling the algebraic identity $$F(x^{n+m})=F(x^m) x^n+ x^m d(x^n)$$ for each $x$ in $R.$ Then $F$ will be a generalized derivation having $d$ as an associated derivation on $R$. On the other hand, in this article, it is also derived that $f$ is a generalized left derivation having a linked left derivation $\delta$ on $R$ if they satisfy the algebraic identity $$f(x^{n+m})=x^n f(x^m)+ x^m \delta(x^n)$$ for each $x$ in $R$ and $k\in \{2, m, n, (n+m-1)!\}$ and at last an application on Banach algebra is presented.

Author Biographies

A. Z. Ansari, Department of Mathematics Faculty of Science Islamic University in Madinah, K.S.A Madinah, India

Department of Mathematics
Faculty of Science Islamic University in Madinah, K.S.A
Madinah, India

N. Rehman, Department of Mathematics, Faculty of Science Aligarh Muslim University, Aligarh, India

Department of Mathematics, Faculty of Science Aligarh Muslim University, Aligarh, India

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