Let $R$ be a ring with unity (not necessarily commutative) and $I$ an ideal of $R$.
Suppose that for every element $a \in I$ there exists an element $c\in I$ such that $ac=a$.
Note that $c$ is related to $a$. Now we have the following question:
Can we say that for every element $a\in I$ there exists an idempotent element $c\in I$ such that $ac=a$?
Of course we have many examples such that the answer is true for them but in general we don’t know.
Consider the ring $R$ of continuous functions $\mathbb R\to\mathbb R$ with compact support.
There are no non-zero idempotents in this ring, yet your condition holds. Indeed, if $a\in R$, let $c\in R$ be any function which is equal to $1$ on the support of $a$.
Later This ring does not have a unit, and you wwanted it to have one. But if $R$ does have a unit then your question is trivial: you can always take $c=1$!