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.

Also see

 * Equivalence of Definitions of Product of Ideals of Ring
 * Definition:Sum of Ideals of Ring

Generalization

 * Definition:Product of Fractional Ideals