# comaximal(coprime)

• Mar 15th 2010, 09:28 PM
KaKa
comaximal(coprime)
Let $A$ be a commutative ring with unity.
If $M,N$ are distinct maximal ideals of $A$, then
(1) $M+N=A$.
(2) $M^a+N^b=A (a,b\ge1)$.
• Mar 15th 2010, 11:45 PM
aliceinwonderland
Quote:

Originally Posted by KaKa
Let $A$ be a commutative ring with unity.
If $M,N$ are distinct maximal ideals of $A$, then
(1) $M+N=A$.
(2) $M^a+N^b=A (a,b\ge1)$.

For (1), the sum of ideals is again an ideal (link). Thus, M+N is an ideal containing M. By hypthesis, M+N should properly contain an maximal ideal M. Thus M+N=A.

For (2), every proper ideal in A is contained in a maximal ideal in A and note that A contains the unity (link).
Assume $M^a+N^b , a,b\ge1$ is a proper ideal in A. Then, $M^a+N^b ,a,b\ge1$ should be contained in a maximal ideal. It follows that $M^a+N^b , a,b\ge1$ should be contained in either M or N (check their intersection). Contradiction !
Thus, $M^a+N^b , a,b\ge1$ is an ideal in A which is not a proper ideal in A. We conclude that $M^a+N^b=A ,a,b\ge1$.