Let be a nonempty index set and let be a ring for each . Prove that the direct product is a ring under componentwise addition and multiplication.
We already know (by this previous exercise) that is an abelian group under componentwise addition, so it suffices to show that multiplication is associative and distributes over addition on both sides.
Let . Then , so that multiplication is associative.
Moreover, . Thus multiplication distributes over addition on the left; distributivity on the right is analogous.
Thus is a ring under componentwise addition and multiplication.