I'm currently working on the following problem:
Let $R$ be a commutative ring, and let $I, J$ be maximal ideals of $R$. Prove that, if $a, b \in R$, there exists some element $c \in R$ such that $c-a \in I$ and $c – b \in J$.
I know that if $I,J$ are maximal ideals of $R$, then for all ideals $K$ of $R$, we have the following:
1) If $I \subseteq K \subseteq R$, either $K = I$ or $K = R$.
2) If $J \subseteq K \subseteq R$, either $K = J$ or $K = R$.
I suppose I'm not seeing the trick to producing this element $c$ such that both $c-a$ and $c-b$ lie in $I$ and $J$, respectively, for two ring elements in $R$. Any help as to producing this element $c$ would be appreciated.
Thanks!
Best Answer
$I+J=R$, since they are maximal, therefore there are $i \in I, j \in J$ such that $i+j=1$.
Let $c=ai+bj$.
$c-a = a(i-1)+bj=a(-j)+bj=(b-a)j \in J\\ c-b = ai+b(j-1)=ai+b(-i)=(a-b)i \in I.$