Given the commutative ring R with unity and $a,b,c \in R$, prove that if the following are coprime principal ideals
$$
⟨a⟩=\{\ ar\ |\ r∈R\ \}\ and\ ⟨b⟩=\{\ br\ |\ r∈R\ \}
$$
and $a\ |\ bc$ (a divides bc), then $a\ |\ c$ (a divides c).
I know that I have to show that a and b are "coprime elements".
For example in $ \mathbb{Z}$, the ideals $m \mathbb{Z}$ and $n \mathbb{Z}$ are coprime ideals when $ \mathbb{Z} = m\mathbb{Z}+ n\mathbb{Z} = GCD(m,n) \mathbb{Z}$ and $GCD(m,n) = \pm1$. So I understand the concept, but when the ring is arbitrary, I'm not sure how can I prove the statement.
Best Answer
It's the same as in the case with $\mathbb Z$. If $I$ and $J$ are coprime ideals of a ring $R$, this means that $I+J = R$. So in particular, you can write $1 = i+j$ for $i \in I, j\in J$. Write out what this means for your ideals and a proof will be readily apparent.