Research article

Involution on prime rings with endomorphisms

  • Received: 08 November 2019 Accepted: 16 February 2020 Published: 27 March 2020
  • MSC : 16N60, 16W10, 16W25

  • Let $\mathcal{R}$ be a prime ring with involution $'*'$ and $\psi: \mathcal{R} \rightarrow \mathcal{R}$ be an endomorphism on $\mathcal{R}$. In this article, we study the action of involution $'*', $ and the effect of endomorphism $\psi$ satisfying $[\psi(x), \psi(x^*)]-[x, x^*]\in \mathcal{Z}(\mathcal{R})$ for all $x\in \mathcal{R}$. In particular, we prove that any centralizing involution on prime rings with involution of characteristic different from two is of the first kind or $\mathcal{R}$ satisfies $s_4$, the standard polynomial identity in four variables. Further, we establish that if a prime ring $\mathcal{R}$ with involution of characteristic different from two admits a non-trivial endomorphism $\psi$ such that $[\psi(x), \psi(x^*)]-[x, x^*]\in \mathcal{Z}(\mathcal{R})$ for all $x\in \mathcal{R}$, then the involution is of the first kind or $\mathcal{R}$ satisfies $s_4$ and $[\psi(x), x] = 0$ for all $x\in \mathcal{R}$.

    Citation: Abdul Nadim Khan, Shakir Ali. Involution on prime rings with endomorphisms[J]. AIMS Mathematics, 2020, 5(4): 3274-3283. doi: 10.3934/math.2020210

    Related Papers:

  • Let $\mathcal{R}$ be a prime ring with involution $'*'$ and $\psi: \mathcal{R} \rightarrow \mathcal{R}$ be an endomorphism on $\mathcal{R}$. In this article, we study the action of involution $'*', $ and the effect of endomorphism $\psi$ satisfying $[\psi(x), \psi(x^*)]-[x, x^*]\in \mathcal{Z}(\mathcal{R})$ for all $x\in \mathcal{R}$. In particular, we prove that any centralizing involution on prime rings with involution of characteristic different from two is of the first kind or $\mathcal{R}$ satisfies $s_4$, the standard polynomial identity in four variables. Further, we establish that if a prime ring $\mathcal{R}$ with involution of characteristic different from two admits a non-trivial endomorphism $\psi$ such that $[\psi(x), \psi(x^*)]-[x, x^*]\in \mathcal{Z}(\mathcal{R})$ for all $x\in \mathcal{R}$, then the involution is of the first kind or $\mathcal{R}$ satisfies $s_4$ and $[\psi(x), x] = 0$ for all $x\in \mathcal{R}$.


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