TITLE: Defining an ideal in the tensor algebra
QUESTION [8 upvotes]: In the wikipedia article about exterior algebra:

The exterior algebra $Λ(V)$ over a vector space $V$ over a field  $K$ is defined as the Quotient algebra of the tensor algebra by the two-sided Ideal $I$ generated by all elements of the form $x \otimes x$ such that $x \in V$.

I want to show that $I$ is an ideal of $T(V)$ but i was confused about the elements of $I$, i mean what is exactly $I$ as a set and more particularly as a subset of $T(V)$.

REPLY [5 votes]: egreg says:

I'm afraid there's not much more that can be said, even in the finite dimensional case

Actually a fairly precise characterization of the ideal generated by the tensors of the form $x \otimes x$ is possible. The ideal is spanned, as a subspace, by tensors of the form
$$x_1 \otimes \dots \otimes x_n \in V^{\otimes n}$$
where the vectors $\{ x_1, \dots x_n \}$ are linearly dependent! This is a nice exercise. It implies, among other things, that a wedge product $x_1 \wedge \dots \wedge x_n \in \wedge^n(V)$ in the exterior algebra vanishes iff the vectors $\{ x_1, \dots x_n \}$ are linearly dependent, which is a standard and useful feature of the exterior algebra.