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

Let be a commutative ring with 1 and let and be -algebras via the ring homomorphisms and . Assuming that the multiplication is well-defined, show that this operation makes into an -algebra.

We already know that 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 is a ring, it suffices to show that multiplication is associative. To that end, let and .

Note that , so that multiplication is associative.

Now note that , so that in fact is a ring with 1. To make into an -algebra, we need a unital ring homomorphism whose image is contained in the center of . To that end, define . Since and , is a ring homomorphism. Moreover, , so that is a unital homomorphism. Finally, note that , so that is contained in the center of . Thus via , is an -algebra.

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