Sum of ideals

The ideal consisting of all sums of an element from each of two ideals.

Sum of ideals

Given ideals $I,J$ in a ring $R$ , their sum is

I+J=\{\,i+j : i\in I,\ j\in J\,\}.

It is the smallest ideal of $R$ containing both $I$ and $J$ .

The sum interacts with quotients via the second isomorphism theorem and measures “comaximality” when $I+J=R$ . In $\mathbb Z$ , sums of ideals encode the gcd .

Examples:

In $\mathbb Z$ , $(m)+(n)=(\gcd(m,n))$ .
In $k[x,y]$ , $(x)+(y)=(x,y)$ .
If $I\subseteq J$ , then $I+J=J$ .