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A ring $R$ is said to be prime if $xRy=0 \implies x=0$ or $y=0.$

Let $R$ be a prime ring and $x \in Z(R)$. Then the ideal generated by $x$ is central i. e. $\langle x \rangle \subseteq Z(R).$

I have seen that this statement is used in some reserch article, but I doubt, whether this question is correct? I think we can find distinct $r_1, r_2\in R$ such that $r_1r_2x^2\neq r_2r_1x^2.$

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    $\begingroup$ Could you link the article, please? It is trivially incorrect for $x = 1$. $\endgroup$ Commented Jun 11 at 8:11
  • $\begingroup$ Sorry, I have done some mistakes while reading the article. $\endgroup$
    – MANI
    Commented Jun 11 at 10:02

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