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My question: What is the coproduct in the category of commutative rings?

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    $\begingroup$ With or without unity? $\endgroup$
    – user239203
    Sep 10 '19 at 21:57
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I will assume that you require your rings to be unital. Then the coproduct will just be the tensor product over $\mathbb{Z}$, i.e. the tensor product as $\mathbb{Z}$-modules (or equivalently abelian groups) with the induced ring structure.

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