## The direct product of rings is a ring under pointwise addition and multiplication

Let $I$ be a nonempty index set and let $R_i$ be a ring for each $i \in I$. Prove that the direct product $\prod_I R_i$ is a ring under componentwise addition and multiplication.

We already know (by this previous exercise) that $\prod_I R_i$ is an abelian group under componentwise addition, so it suffices to show that multiplication is associative and distributes over addition on both sides.

Let $(\prod a_i), (\prod b_i), (\prod c_i) \in \prod_I R_i$. Then $(\prod a_i)((\prod b_i)(\prod c_i)) = (\prod a_i)(\prod (b_ic_i)) = \prod a_i(b_ic_i)$ $= \prod (a_ib_i)c_i$ $= (\prod b_ic_i)(\prod a_i)$ $= ((\prod a_i)(\prod b_i))(\prod c_i)$, so that multiplication is associative.

Moreover, $(\prod a_i)((\prod b_i) + (\prod c_i)) = (\prod a_i)(\prod (b_i + c_i))$ $= \prod a_i(b_i + c_i)$ $= \prod (a_ib_i + a_ic_i)$ $= (\prod a_ib_i) + (\prod a_ic_i)$ $= (\prod a_i)(\prod b_i) + (\prod a_i)(\prod c_i)$. Thus multiplication distributes over addition on the left; distributivity on the right is analogous.

Thus $\prod_I R_i$ is a ring under componentwise addition and multiplication.