Let R be acommutative ring and let b be a fixed element of R. Prove that the set b = {r is in R and r = cb for some element c in R} is an ideal of R.
Define $\displaystyle (b) = \{ rb | r\in R\}$. To show $\displaystyle (b)$ is an ideal of $\displaystyle R$ you need to show if $\displaystyle x,y\in (b)$ then $\displaystyle x\pm y \in (b)$, and also that if $\displaystyle r\in R$ then $\displaystyle r(b),(b)r \subseteq (b)$. These should be clear by definition.