Verify that the tensor product of two R-algebras is an R-algebra

Let R be a commutative ring with 1 and let A and B be R-algebras via the ring homomorphisms \alpha : R \rightarrow A and \beta : R \rightarrow B. Assuming that the multiplication (a_1 \otimes b_1)(a_2 \otimes b_2) = a_1a_2 \otimes b_1b_2 is well-defined, show that this operation makes A \otimes_R B into an R-algebra.

We already know that A \otimes_R B is an abelian group with respect to the usual addition. Moreover, in our proof (not given here) that this multiplication is well-defined, we found that multiplication is bilinear- that is, it distributes over addition from both sides. To show that A \otimes_R B is a ring, it suffices to show that multiplication is associative. To that end, let a_1, a_2, a_3 \in A and b_1, b_2, b_3 \in B.

Note that (a_1 \otimes b_1)((a_2 \otimes b_2)(a_3 \otimes b_3)) = (a_1 \otimes b_1)(a_2a_3 \otimes b_2b_3) = a_1(a_2a_3) \otimes b_1(b_2b_3) = (a_1a_2)a_3 \otimes (b_1b_2)b_3 = (a_1a_2 \otimes b_1b_2)(a_3 \otimes b_3) = ((a_1 \otimes b_1)(a_2 \otimes b_2))(a_3 \otimes b_3), so that multiplication is associative.

Now note that (1 \otimes 1)(a \otimes b) = a \otimes b = (a \otimes b)(1 \otimes 1), so that in fact A \otimes_R B is a ring with 1. To make A \otimes_R B into an R-algebra, we need a unital ring homomorphism \gamma : R \rightarrow A \otimes_R B whose image is contained in the center of A \otimes_R B. To that end, define \gamma(r) = \alpha(r) \otimes 1. Since \gamma(r+s) = \alpha(r+s) \otimes 1 = (\alpha(r) + \alpha(s)) \otimes 1 = \alpha(r) \otimes 1 + \alpha(s) \otimes 1 = \gamma(r) + \gamma(s) and \gamma(rs) = \alpha(rs) \otimes 1 = \alpha(r)\alpha(s) \otimes 1 = (\alpha(r) \otimes 1)(\alpha(s) \otimes 1) = \gamma(r)\gamma(s), \gamma is a ring homomorphism. Moreover, \gamma(1) = \alpha(1) \otimes 1 = 1 \otimes 1, so that \gamma is a unital homomorphism. Finally, note that \gamma(r)(a \otimes b) = (\alpha(r) \otimes 1)(a \otimes b) = \alpha(r)a \otimes b = a \alpha(r) \otimes b = (a \otimes b)(\alpha(r) \otimes 1), so that \mathsf{im}\ \gamma is contained in the center of A \otimes_R B. Thus via \gamma, A \otimes_R B is an R-algebra.

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