I need to confirm that I have understood the definition of the exterior algebra of a vector space correctly. I would be very grateful if someone could confirm that this is correct, or point out any issues.
Let $V$ be a vector space over a field $K$. Then the tensor algebra $$T(V):=K\oplus V\oplus (V\otimes V)\oplus(V\otimes V\otimes V)\oplus\cdots$$ has elements which are finite sums of elementary tensors $x_1\otimes\cdots\otimes x_n$, where $x_i\in V$ (including elements of $K$ for the '$0$-tensors').
Then we obtain the exterior algebra $\bigwedge(V)$ by setting $x\otimes x=0$ for all $x\in V$ (within $T(V)$) and writing the image of $x_1\otimes\cdots\otimes x_n$ under the quotient by the ideal generated by elements of the form $x\otimes x$ as $x_1\wedge\cdots\wedge x_n$ (so that $x\wedge x=0$ for all $x\in V$).
Any issues here?