Smallest Ideal containing $a$ in commutative ring without unity Let  $R$ be a commutative ring without unity such that $a$ belongs to $R$. Describe the smallest ideal containing $a$.
We know that in a commutative ring with unity $(a) = ${$r.a$ | $r$ belongs to $R$} but what is the case without unity.
I think $(a)$ = {$r.a+ n.a$ | $r$ in $R$ and $n$ belongs to integers}
but how would we prove this if it is correct.
Thanks for help.

The path is clear: you show the set you describe is an ideal, and then you explain n why every ideal containing $a$ must contain your set. All of this follows from axiom checking.
