Let $R$ be a commutative ring. Take $a,b \in R$, if $Ra + Rb = R$, show that $Ra \cap Rb \subset Rab$.

Question

Let $R$ be a commutative ring. Take $a,b \in R$; if $Ra + Rb = R$, I want to show that $Ra \cap Rb \subset Rab$.

Thoughts: Take $r \in Ra \cap Rb$, then $r = r_1a + r_2b$ for some $r_1, r_2 \in R$, but it is also true that $r = r_3a = r_4b$ for some $r_3, r_4 \in R$.

This is about as far as I've got, hints appreciated.

Answer 1

By assumption there exist some $x,y\in R$ with $1=ax+by$. If $r\in Ra\cap Rb$, then $r=rax+rby$ and you can check that $rax$ and $rby$ are both elements of $Rab$.