# Definition:Product of Ideals of Ring

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## 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

### Generalization

## Sources

- 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 6$: Rings and fields