If $R$ is commutative ring with identity, prove that $R$ is and integral domain iff cancelation holds in $R$

Question

Please check my proof

$\Leftarrow $ Suppose cancelation holds in $R$, but $R$ is not integral domain. There must exist zero divisor in $R$ We supoose $a$ is zero divisor and $b\neq c$ then suppose

$ab=ac$

cancel a form both side $b=c$ that is false then the if cancellation hold in $R$, $R$ can't has zero divisor or it's integral domain

$\Rightarrow $ Because $R$ is integral domain, there are no zero divisor in $R$

Consider $c=d$

$ac=ad$

cancel $a$ from both sides

$c=d$

that is true then cancellation holds if it is integral domain

Answer 1

Take $a \in R$ and consider $\mu : R \to R$ given by $\mu(x)=ax$.

Then $\mu$ is a homomorphism of the additive group of $R$.

By definition, $a$ is a zero divisor iff $\ker\mu \ne 0$. Therefore, $R$ is a domain iff $\ker\mu = 0$ for all $a\ne0$.

It is well known that $\ker\mu = 0$ is equivalent to $\mu$ being injective, which is just another way of expressing cancellation.

Answer 2

Ideas:

$\;\Longleftarrow\;$ Suppose cancelation holds but $\;a\neq0\;$ is a zero divisor. Then there exists $\;b\neq0\;$ such that

$$0=a\cdot0=a\cdot b\;\;\ldots\text{and now finish}$$

$\;\implies\;$ Supose $\;R\;$ is an ID, and also that $\;a\neq0\;$ , so if

$$ab=ac\implies a(b-c)=0\;,\;\;\text{but...}$$