|home | alphabetical index|
Tensor product of R-algebrasIn 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.
|copyright © 2004 FactsAbout.com|