When is $(r_1, ..., r_k) = r_1R + ... + r_kR$?

Question

Let $R$ be a ring and $(r_1, ..., r_k)$ be the ideal generated by $\{r_1, ..., r_k\}$. Does $R$ need to have unit for $(r_1, ..., r_k) = r_1R + ... + r_kR$ to hold?

Answer 1

I post my comment as an answer:

We look at the easiest case, $k = 1$. In this case the equality $(r) = rR$ is true if and only if there is some $x \in R$ with $rx = r$. So it suffices to find a ring $R$ and an element $r \in R$ such that $rx \neq r$ for all $x \in R$.

As an example we take $R = 2\mathbb{Z}$ and $r = 2 \in 2\mathbb{Z}$. Then $(2) = 2 \mathbb{Z} \subsetneq 4 \mathbb{Z} = 2(2\mathbb{Z})$.