Let $R$ be a ring. An endomorphism $\alpha:R\to R$ is said to be right central reflexive if for all $a,b\in R,$ $aRb=(0)\implies bR\alpha (a)\subset Z(R)$, where $Z(R)$ denotes the centre of the ring.
The ring $R$ is called right $\alpha$-central reflexive if $\alpha$ is a right central endomorphism for $R$.
Let us consider a statement as follows:
For all $a,b\in R$, $aR\alpha (b)=(0)\implies bRa\subset Z(R)$.....(*)
It can be proved that if $\alpha:R\to R$ is onto, $R$ is right central $\alpha$-reflexive implies (*) is true. I want to show show that surjectivity of $\alpha$ is not superfluous. But I could not find an into endomorphism $\alpha$ such that $(*)$ does not hold but $R$ is a right central $\alpha$-reflexive ring.
Please help me in this regard.