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!

An integral domain is defined to be a ring with the cancellation property. So cancellation does not work in general in a ring that is not an integral domain. Zero divisors, which exist in any ring that is not an integral domain, will provide counterexamples. Consider $R^2$ as a direct product of any unital ring $R$ with itself. Addition and multiplication are defined component-wise as in $R$. Take  $p=(a,0)$, for any nonzero a $   \epsilon $ $R$. Then take $c=(1,b)$ for any b $\epsilon$ $R$. Then $pc=(a,0)(1,b)=(a,0)=p$. So no, it does not hold. It does most certainly hold in integral domains, however.
