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Because of the generality of the tensor algebra, many other algebras of interest are constructed by starting with the tensor algebra and then imposing certain relations on the generators, i.e. by constructing certain quotients of T(V). Examples of this are the exterior algebra, Clifford algebras and universal enveloping algebras.

The formal construction of T(V) is as a direct sum of graded parts T^k for k = 0,1,2, ... ; where T^k is the tensor product of V with itself k times, and T⁰ is K as one-dimensional vector space.

The multiplication map on Tⁱ and T^j to T^i+j is the natural juxtaposition on pure tensors, extended by bilinearity. That is, the tensor algebra contains all covariant tensors on V, of any rank.

One can also refer to T(V) as the free algebra on the vector space V. In fact, T is a functorFor the usage in computer science, see the function object article. In category theory, a functor is a special type of mapping between categories. Functors can be thought of as morphisms in the category of all ( small) categories. Functors were first cons from the category of K-vector spaces to the category of unital associative K-algebras, and it is left adjoint to the functor taking any unital associative K-algebra to its underlying vector space. Spelled out, this translates into the following universal propertyIn category theory, abstract algebra and other fields of mathematics, frequently constructions are defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. These properties are called uni of T(V): any K-linear mapIn mathematics, a linear transformation (also called linear operator or linear map is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it "prese from V to some unital associative K-algebra A can be uniquely extended to a unital algebra homomorphism from T(V) to A.

The construction generalises straightforwardly to the tensor algebra of any moduleAlgebra In abstract algebra, a module is a generalization of a vector space. In a vector space the set of scalars forms a field whereas in a module the scalars just form a ring. Much of the theory of modules consists of recovering desirable properties of M over a commutative ringIn ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. This means that if a and b are any elements of the ring, and if the multiplication operation is written as then a '. If R is a non-commutative ringIn ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers. History See Ring theory Definition and notation A ring is an a, one can still perform the construction for any R-R bimoduleIn abstract algebra a bimodule is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible. Formally, if R and S are two rings, then an R ''S bimodule is an abelian group M such that: # M is a le M. (It doesn't work for ordinary R-modules because the iterated tensor products cannot be formed.)