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.

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