Question

Show that a nonzero central element of a prime ring R is not a zero-divisor in R.

Answer 1

Question 1.

Show that a nonzero central element of a prime ring $R$ is not a zero-divisor in $R$.

Solution.

Recall that a ring is called prime if it is non-zero and for any $a,b\in R$ the equality $aRb=0$, implies $a=0$ or $b=0$. Suppose $a$ is a central element of $R$ and $ab=0$ for some $b\in R$. Then for any $c\in R$ we have

$acb=(ac)b=(ca)b=c(ab)=c\cdot 0=0.$

So, $aRb=0$ and hence $a=0$ or $b=0$, since $R$ is prime. Thus, $a$ is not a divisor of zero. ∎