Associative rings and algebras Definition

Product of rings

Builds on Ring

Definition

Let $I$ be a set and let $(R_i)_{i \in I}$ be a family of rings. The product $\prod_{i \in I} R_i$ is the set of families $(a_i)_{i \in I}$ with $a_i \in R_i$ for each $i \in I$ . Its addition is $(a_i)_{i \in I} + (b_i)_{i \in I} = (a_i + b_i)_{i \in I}$ . Its multiplication is $(a_i)_{i \in I} \cdot (b_i)_{i \in I} = (a_i \cdot b_i)_{i \in I}$ . With these operations, $\prod_{i \in I} R_i$ is a ring, with zero $(0)_{i \in I}$ and identity $(1)_{i \in I}$ .

Remarks

For two rings $R$ and $S$ , the product is $R \times S = \{(r, s) : r \in R, s \in S\}$ . For each $j \in I$ , the projection $\prod_{i \in I} R_i \to R_j$ , $(a_i)_{i \in I} \mapsto a_j$ , is a ring homomorphism. The product of two nonzero rings is not an integral domain: in $R \times S$ , $(1, 0) \cdot (0, 1) = (0, 0)$ .

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Group concept
Ring definition
Product of rings definition

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Tag	`0MNZCN4BQK`
URI	`m2://0MNZCN4BQK`
Kind	definition
Writing	W1 stated
MSC	16B99
Defines	Product of rings
Uses	Ring

Product of rings

Definition

Remarks

Reading path

Where this leads

Further reading

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