Product of ideals

Given ideals $I,J$ in a ring $R$, their product $IJ$ is the ideal generated by the set $\{ij:i\in I,\ j\in J\}$. Equivalently, $IJ$ consists of all finite sums $\sum_{k=1}^n i_k j_k$ with $i_k\in I$ and $j_k\in J$.

One always has $IJ\subseteq I\cap J$ and $IJ\subseteq I$ and $IJ\subseteq J$; equality $IJ=I\cap J$ is a strong condition (e.g. it holds for comaximal ideals by the Chinese remainder theorem framework). In commutative algebra, products model multiplicative behavior of vanishing conditions.

Examples:

• In $\mathbb Z$, $(m)(n)=(mn)$.
• In $k[x]$, $(x)(x)=(x^2)$.
• In $k[x,y]$, $(x)(y)=(xy)\subseteq (x)\cap (y)$ (in fact equality holds here).