Definition:Product of Ideals of Ring

Definition

Let $\left({R, +, \circ}\right)$ be a commutative ring.

Let $I,J$ be ideals of $R$.

Definition 1

The product of $I$ and $J$ is the set of all finite sums:

$IJ = \{a_1 b_1 + \cdots + a_r b_r : a_i \in I, b_i \in J, r \in \N \}$

Definition 2

The product of $I$ and $J$ is the ideal generated by their product as subsets.