Let R be a commutative ring with unity in which the cancellation law for multiplication holds. That is, if a, b, and c are elements of R, then a does not equal to 0 and ab = ac always imply b = c. Prove that R is an integral domain.

Here is my work:

since a can't be zero, then b = c is true. Since R is a commutative ring with unity, then eb = be = b and ec = ce = c which implies that b and c can't be zero. If not, then eb = be = 0 and ec = ce = 0. Therefore, R is an integral domain.

Am I right?