tensor algebra
The tensor algebra is the most permissive algebra you can build from a vector space: it lets you multiply vectors freely, imposing no relations whatsoever except those forced by being an associative algebra. Think of vectors as letters and the product as concatenation — a 'word' is just a string of vectors, and you add formal combinations of words. Nothing commutes, nothing collapses; it is the free associative algebra on the space.
Given an R-module V, set T^0(V) = R, T^1(V) = V, and T^k(V) = V ⊗ ... ⊗ V (k factors). The tensor algebra is the direct sum T(V) = ⊕_{k≥0} T^k(V), with multiplication given by concatenation of tensors, (v_1 ⊗ ... ⊗ v_p)·(w_1 ⊗ ... ⊗ w_q) = v_1 ⊗ ... ⊗ v_p ⊗ w_1 ⊗ ... ⊗ w_q. This makes T(V) a graded associative R-algebra with unit 1 in T^0.
Its defining feature is a universal property: any R-linear map V -> A into an associative R-algebra A extends uniquely to an algebra homomorphism T(V) -> A. Thus T(V) is free on V in the category of algebras. The exterior algebra and the symmetric algebra are both quotients of T(V) — by the ideal generated by v ⊗ v, respectively by v ⊗ w − w ⊗ v — so the tensor algebra is the common parent from which the others are carved.
If dim V = n, then dim T^k(V) = n^k, so the graded pieces of T(V) have dimensions 1, n, n^2, n^3, ... . For n = 1, T(V) ≅ R[x], the polynomial ring in one variable (here the single variable commutes trivially with itself).
Dimensions grow like a geometric ladder n^k; one dimension recovers a polynomial ring.