Editing Tensor algebra (section)

==Construction==
Let ''V'' be a [[vector space]] over a [[field (mathematics)|field]] ''K''. For any nonnegative [[integer]] ''k'', we define the '''''k''th tensor power''' of ''V'' to be the [[tensor product]] of ''V'' with itself ''k'' times:
:<math>T^kV = V^{\otimes k} = V\otimes V \otimes \cdots \otimes V.</math>
That is, ''T''<sup>''k''</sup>''V'' consists of all tensors on ''V'' of [[tensor order|order]] ''k''. By convention ''T''<sup>0</sup>''V'' is the [[ground field]] ''K'' (as a one-dimensional vector space over itself).

We then construct ''T''(''V'') as the [[direct sum of vector spaces|direct sum]] of ''T''<sup>''k''</sup>''V'' for ''k'' = 0,1,2,…
:<math>T(V)= \bigoplus_{k=0}^\infty T^kV = K\oplus V \oplus (V\otimes V) \oplus (V\otimes V\otimes V) \oplus \cdots.</math>
The multiplication in ''T''(''V'') is determined by the canonical isomorphism
:<math>T^kV \otimes T^\ell V \to T^{k + \ell}V</math>
given by the tensor product, which is then extended by linearity to all of ''T''(''V''). This multiplication rule implies that the tensor algebra ''T''(''V'') is naturally a [[graded algebra]] with ''T''<sup>''k''</sup>''V'' serving as the grade-''k'' subspace. This grading can be extended to a '''Z'''-grading by appending subspaces <math>T^{k}V=\{0\}</math> for negative integers ''k''.

The construction generalizes in a straightforward manner to the tensor algebra of any [[module (mathematics)|module]] ''M'' over a [[commutative ring|''commutative'' ring]]. If ''R'' is a [[non-commutative ring]], one can still perform the construction for any ''R''-''R'' [[bimodule]] ''M''. (It does not work for ordinary ''R''-modules because the iterated tensor products cannot be formed.)