For a commutative ring R without identity, there exists a∈R such that Ra≠R

Question

Is this statement true? Then how to prove it?

For a non trivial commutative ring $R$ without identity, there exists $a \in R\setminus \{0\}$ such that $Ra \not = R$

Answer 1

Let $R$ be a commutative ring without identity, and assume $Ra = R$ for some fixed $a$. Then in particular, there exists $e \in R$ with $ea=a$. For all $r \in R$, $e(ra) = (ea) r = ar = ra$, and since every element of $R$ can be written as $ra$ for some $r \in R$, this proves $e$ is an identity for $R$.

So this proves the stronger result that if $R$ is commutative and $Ra = R$ for any $a \in R$, then $R$ has an identity element.

Answer 2

In the zero ring, $Ra=R$ for all $a$.

In a nonzero ring, $R\cdot 0\neq R$.

Edit: The question has since changed. The answer is still "not necessarily". In a field, $Ra=R$ for every nonzero $a$.