Assume we have a non-commutative unital ring $R$ and an element $r$ not in the center. Define a map $$\phi_r:R\rightarrow R$$ $$x\mapsto rx$$ Can this ever be a ring homomorphism?
If it can be then $r$ has to be idempotent, and the kernel (the set of $r$'s zero divisors with $0$) must be an ideal. I'm pretty sure that this can't be a homomorphism, but I can't prove it. Although I can't promise that I just haven't been clever enough to think of an example where this works. Some simple proof showing that $r$ has to be in the center is what I've striven for, but alas…
Edit: Suppose I should add that I'm adopting the definition of a ring homomorphism that isn't necessarily (multiplicative) identity invariant.
Best Answer
Let $R$ be the ring of $2\times 2$ upper triangular matrices with coefficients in some field, and let $r=\begin{pmatrix}0&0\\0&1\end{pmatrix}$.