Intersection of ideals

Given ideals $I,J$ in a ring $R$ , their intersection is the set-theoretic intersection $I\cap J=\{x\in R:x\in I\text{ and }x\in J\}$ .

Remarks

The intersection $I\cap J$ is again an ideal, and it is the largest ideal contained in both $I$ and $J$ . Intersections appear naturally in primary decomposition and in comparing congruence conditions.

Examples

In $\mathbb Z$ , $(m)\cap (n)=(\operatorname{lcm}(m,n))$ , where $\operatorname{lcm}$ is the lcm.
In $k[x,y]$ , $(x)\cap (y)=(xy)$ .
If $I\subseteq J$ , then $I\cap J=I$ .