mathematics, the tensor product of two algebras over a commutative ring R is also an R-algebra. This gives the tensor product of algebras. When the ring...

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....

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...

abstract algebra. The tensor product of an algebra and a module can be used for extension of scalars. For a commutative ring, the tensor product of modules...

R-algebras. Tensor products The tensor product of two R-algebras is also an R-algebra in a natural way. See tensor product of algebras for more details...

In mathematics, the tensor product of two fields is their tensor product as algebras over a common subfield. If no subfield is explicitly specified, the...

category of commutative algebras over a commutative ring is a tensor product of algebras. A coproduct in the category of algebras is a free product of algebras...

glossary of tensor theory. For expositions of tensor theory from different points of view, see: Tensor Tensor (intrinsic definition) Application of tensor theory...

analysis, the tensor product of Hilbert spaces is a way to extend the tensor product construction so that the result of taking a tensor product of two Hilbert...

mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map...

Tensor product of Hilbert spaces, endowed with a special inner product as to remain a Hilbert space Other topological tensor products Tensor product of...

that the Weyl algebra is a quantization of the symmetric algebra. Weyl algebras and Clifford algebras admit a further structure of a *-algebra, and can be...

}(V).} In particular, the exterior algebra of a direct sum is isomorphic to the tensor product of the exterior algebras: ⋀ ( V ⊕ W ) ≅ ⋀ ( V ) ⊗ ⋀ ( W )...

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...

algebra over K that splits over L and such that deg A = [L : K] arises in this way. The tensor product of algebras corresponds to multiplication of the...

The outer product of tensors is also referred to as their tensor product, and can be used to define the tensor algebra. The outer product contrasts with:...

product Outer product Kronecker delta Levi-Civita symbol Multilinear form Pseudoscalar Pseudovector Spinor Tensor Tensor algebra, Free algebra Tensor...

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...

space tensor product of two Hilbert spaces is the completion of their algebraic tensor product. One can define a tensor product of von Neumann algebras (a...

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...

K and addition is induced by the tensor product of algebras. It arose out of attempts to classify division algebras over a field and is named after the...

define a free product of two algebras. Let A and B be algebras over a commutative ring R. Consider their tensor algebra, the direct sum of all possible...

algebras of pseudo-differential operators acting on spaces of distributions), the term operator algebra is usually used in reference to algebras of bounded...

the category of R-algebras to the category of sets. Free algebras over division rings are free ideal rings. Cofree coalgebra Tensor algebra Free object...

In mathematics, tensor calculus, tensor analysis, or Ricci calculus is an extension of vector calculus to tensor fields (tensors that may vary over a manifold...

of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras over K, with addition given by the tensor product...

objects (also called quotient algebras in universal algebra, and cokernels in category theory). For many types of algebraic structure, the fundamental theorem...

specifically multilinear algebra, a dyadic or dyadic tensor is a second order tensor, written in a notation that fits in with vector algebra[disambiguation needed]...

with a group operation given by the tensor product of algebras. The resulting group is called the Brauer group Br(F) of the field F. It is always a torsion...

Examples include Lie algebras, Jordan algebras, the octonions, and three-dimensional Euclidean space equipped with the cross product operation. Since it...