Main Page | See live article | Alphabetical index 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 RZS 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 ARB 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'b.c'd = acbd, to have a well-defined product on ARB; 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. This article is from Wikipedia. All text is available under the terms of the GNU Free Documentation License.