[Math] When does the cancellation law hold for the ring

abstract-algebraring-theory

Let $R$ be an arbitrary ring. Now, we assume that we don't know whether $R$ has the multiplicative identity or not.

I know that $R$ has no zero divisors if and only if the cancellation law holds. So, suppose $R$ has no zero divisors. Consider for a nonzero element $a\in R$, $ab=a$ for some $b\in R$. Now, I want to apply for the cancellation law, but, if so, we have $b=1$, where $1$ is the multiplicative identity of $R$.

I think it is false because we don't know whether the ring has the unity.

Thus, I'm wondering when the cancellation law holds.

Best Answer

If $ab=a$ and $R$ has no zero divizors, then for any $c\in R$ $$ a(bc) = (ab)c = ac, $$ so $bc = c$. So $R$ does have a unity, and it is $b$.

Upd. $cb = c$ holds as well since $(cb)d = c(bd) = cd$.