Let and be rings. Prove that the direct product is a ring under componentwise addition and multiplication. Prove that is commutative if and only if and are commutative. Prove that has an identity if and only if and have identities.
We already know that is an abelian group under componentwise addition. Now if and , we have . So componentwise multiplication is associative. Moreover, . Thus componentwise multiplication distributes over componentwise addition on the left; similarly, it distributes on the right. Thus is a ring.
If and are commutative, then , so that is commutative. If is commutative, then . Comparing entries, we have and , so that and are commutative.
If and have 1s, then and , so that is an identity. Suppose is an identity. Then for all and , we have . Then and ; by the uniqueness of identities, both and have a 1.