Tensor product of modules information

for extension of scalars. For a commutative ring, the tensor product of modules can be iterated to form the tensor algebra of a module, allowing one to...

v\otimes w} is called the tensor product of v and w. An element of V ⊗ W {\displaystyle V\otimes W} is a tensor, and the tensor product of two vectors is sometimes...

as R-modules, their tensor product A ⊗ R B {\displaystyle A\otimes _{R}B} is also an R-module. The tensor product can be given the structure of a ring...

derived tensor product of M and N. In particular, π 0 ( M ⊗ R L N ) {\displaystyle \pi _{0}(M\otimes _{R}^{L}N)} is the usual tensor product of modules M and...

One-form Tensor product of modules Application of tensor theory in engineering Continuum mechanics Covariant derivative Curvature Diffusion tensor MRI Einstein...

are O-modules, then their tensor product, denoted by F ⊗ O G {\displaystyle F\otimes _{O}G} or F ⊗ G {\displaystyle F\otimes G} , is the O-module that...

R, the category of left modules over R is the category whose objects are all left modules over R and whose morphisms are all module homomorphisms between...

and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used...

differential p-form with values in a vector bundle E. Tensor product of modules To construct a tensor-product bundle over a paracompact base, first note the...

the derived functors of the tensor product of modules over a ring. Along with the Ext functor, Tor is one of the central concepts of homological algebra...

In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the canonical pairing of a vector space and its dual. In components...

associativity can be expressed as follows. By the universal property of a tensor product of modules, the multiplication (the R-bilinear map) corresponds to a unique...

the relevant diagrams commute. The ordinary tensor product makes vector spaces, abelian groups, R-modules, or R-algebras into monoidal categories. Monoidal...

combines several modules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with...

topological tensor product of two topological vector spaces. For Hilbert spaces or nuclear spaces there is a simple well-behaved theory of tensor products (see...

geometry Tor functor, the derived functors of the tensor product of modules over a ring Torsion-free module, in algebra See also Torsion-free (disambiguation)...

flat if taking the tensor product over R with M preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces...

tensor product is in general not necessarily complete, so its completion is called the completed injective tensor products. Injective tensor products...

In functional analysis, an area of mathematics, the projective tensor product of two locally convex topological vector spaces is a natural topological...

In mathematics, the tensor product of representations is a tensor product of vector spaces underlying representations together with the factor-wise group...

object (empty product) is the unit object. The category of bimodules over a ring R is monoidal (using the ordinary tensor product of modules), but not necessarily...

the characteristic of F is a prime number p for which there is some p-torsion in the homology. Consider the tensor product of modules Hi(X; Z) ⊗ A. The...

tensor algebra of a vector space V, denoted T(V) or T•(V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product....

structure, from the tensor algebra. See the article on tensor algebras for a detailed treatment of the topic. The exterior product of multilinear forms...

monoidal product is given by the tensor product of modules and the internal Hom M ⇒ N {\displaystyle M\Rightarrow N} is given by the space of R-linear...