Search results
Results from the WOW.Com Content Network
The tensor product distributes over direct sums in the following sense: if N is some right R-module, then the direct sum of the tensor products of N with M i (which are abelian groups) is naturally isomorphic to the tensor product of N with the direct sum of the M i.
The tensor product of two vector spaces is a vector space that is defined up to an isomorphism.There are several equivalent ways to define it. Most consist of defining explicitly a vector space that is called a tensor product, and, generally, the equivalence proof results almost immediately from the basic properties of the vector spaces that are so defined.
For an arbitrary family of groups indexed by , their direct sum [2] is the subgroup of the direct product that consists of the elements () that have finite support, where by definition, () is said to have finite support if is the identity element of for all but finitely many . [3] The direct sum of an infinite family () of non-trivial groups is ...
The direct sum and direct product are not isomorphic for infinite indices, where the elements of a direct sum are zero for all but for a finite number of entries. They are dual in the sense of category theory: the direct sum is the coproduct, while the direct product is the product.
In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) ... Then M is the direct sum = ...
Ab, the category of abelian groups, with the direct sum of abelian groups as monoidal product and the trivial group as unit. More generally, the category R-Mod of (left) modules over a ring R (commutative or not) becomes a cocartesian monoidal category with the direct sum of modules as tensor product and the trivial module as unit.
This representation is called outer tensor product of the representations and . The existence and uniqueness is a consequence of the properties of the tensor product. Example. We reexamine the example provided for the direct sum:
Functors are often defined by universal properties; examples are the tensor product, the direct sum and direct product of groups or vector spaces, construction of free groups and modules, direct and inverse limits. The concepts of limit and colimit generalize several of the above.