Tensor product of R-algebras
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In mathematics, there is a construction in abstract algebra of the tensor product of commutative rings; which puts a ring structure on the tensor product as abelian groups of two commutative rings R and S. This structure on R<math> \otimes <math>ZS then makes it a coproduct in the category of commutative rings.
More generally, if R is a commutative ring and A and B are commutative R-algebras, we can make A<math> \otimes <math>RB into a commutative R-algebra by the same formula, getting a coproduct in the same way: the previous construction being the case R = Z. We observe the multilinear nature of the product a<math> \otimes <math>b.c<math> \otimes <math>d = ac<math> \otimes <math>bd, to have a well-defined product on A<math> \otimes <math>RB; the ring axioms and R-linearity can be checked too.
This construction is of constant use in algebraic geometry: working in the opposite category to that of commutative R-algebras, it provides pullbacks of affine schemes, otherwise known as fiber products.
See also tensor product of fields.