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...

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

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...

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

product) Tensor product of modules, the same operation slightly generalized to modules over arbitrary rings Kronecker product, the tensor product of matrices...

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...

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...

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...

tensor product of vector bundles E, F (over the same space X) is a vector bundle, denoted by E ⊗ F, whose fiber over each point x ∈ X is the tensor product...

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...

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...

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

module is intuitively a module that has an inverse with respect to the tensor product. Invertible modules form the foundation for the definition of invertible...

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

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...

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...

called flat if taking the tensor product of it with any exact sequence of R-modules preserves exactness. Torsionless A module is called torsionless if...

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)...

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...

{\displaystyle R/I\otimes _{R}R/J\simeq R/(I+J)} (for this identity, see tensor product of modules#Examples.) Example: Let X ⊂ P n {\displaystyle X\subset \mathbb...

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...

a tensor field is a function assigning a tensor to each point of a region of a mathematical space (typically a Euclidean space or manifold) or of the...

In mathematics, the tensor-hom adjunction is that the tensor product − ⊗ X {\displaystyle -\otimes X} and hom-functor Hom ⁡ ( X , − ) {\displaystyle \operatorname...

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...

component-free approach to the theory of a tensor views a tensor as an abstract object, expressing some definite type of multilinear concept. Their properties...

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....

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...

some p {\displaystyle p} -torsion in the homology. Consider the tensor product of modules H i ( X , Z ) ⊗ A {\displaystyle H_{i}(X,\mathbb {Z} )\otimes...

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

multiples of π {\displaystyle \pi } replaced by zero). For some purposes, this definition can be made using the tensor product of modules over Z {\displaystyle...