If $p=pc$, does this imply $c=1$ in ring $R$? Let $R$ be a ring, where $p=pc$ for $p,c\in R$. Does $c$ have to be equal to $1$? $R$ may not be an integral domain. 
Note that $p,c\neq 0$.
Thanks!

Nontrivial idempotents give counterexamples. The simplest one that comes to mind is
$$
\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}
$$
in $M_2(\mathbb{Z})$, for instance.
