Intersection of ideals

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

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 $I\cap J=I$.