Principal Ideal Domain Basics.

Question

Let $R$ be a Principal Ideal Domain and $a,b,c,d$ elements in $R$, such that $ab-cd=1$. I am trying to figure out why $Rb \cap Rd=Rdab+Rbcd$.

In case this is true, I am wondering weather it is enough to claim, that $ab-cd=u$ with $u$ a unit in $R$.

Answer 1

Let $x = mb = nd \in Rb \cap Rd$. Then

$$x = x\cdot 1 = x (ab - cd) = ndab - mbcd \in Rdab + Rbcd.$$

Conversely, $Rdab \subset Rbd \subset Rb \cap Rd$ and $Rbcd \subset Rbd \subset Rb \cap Rd$, so $Rdab + Rbcd \subset Rb \cap Rd$.

It would indeed work with $ab - cd = u$ for any unit $u$, you'd multiply by $u^{-1}$ then at some point.

The gist of it is that $ab - cd = 1$ says that $b$ and $d$ are coprime, as are $a$ and $c$, and that means $Rb \cap Rd = Rbd$ resp. $Ra + Rc = R$.