Construction
The construction of M ⊗ N takes a quotient of a free abelian group with basis the symbols m ⊗ n for m in M and n in N by the subgroup generated by all elements of the form
- −(m+m′) ⊗ n + m ⊗ n + m′ ⊗ n
- −m ⊗ (n+n′) + m ⊗ n + m ⊗ n′
- (m·r) ⊗ n − m ⊗ (r·n)
where m,m′ in M, n,n′ in N, and r in R. The function which takes (m,n) to the coset containing m ⊗ n is bilinear, and the subgroup has been chosen minimally so that this map is bilinear.
The direct product of M and N is rarely isomorphic to the tensor product of M and N. When R is not commutative, then the tensor product requires that M and N be modules on opposite sides, while the direct product requires they be modules on the same side. In all cases the only function from M × N to Z which is both linear and bilinear is the zero map.
Read more about this topic: Tensor Product Of Modules
Famous quotes containing the word construction:
“Striving toward a goal puts a more pleasing construction on our advance toward death.”
—Mason Cooley (b. 1927)
“The construction of life is at present in the power of facts far more than convictions.”
—Walter Benjamin (1892–1940)
“No real “vital” character in fiction is altogether a conscious construction of the author. On the contrary, it may be a sort of parasitic growth upon the author’s personality, developing by internal necessity as much as by external addition.”
—T.S. (Thomas Stearns)