Let $V$ be a vector space over a field $K$ and let $T(V)$ be its tensor algebra; that is, $$T(V)=\displaystyle\bigoplus_{k=0}^{\infty}V^{\otimes k}=K\oplus V\oplus V\oplus(V\otimes V)\oplus(V\otimes V\otimes V)\oplus\cdots\text{.}$$
Thus an element of $T(V)$ looks like $k\oplus v_1\oplus (v_2\otimes v_3)\oplus\cdots$. I would like to define the rank of an arbitrary element in $T(V)$. I am not sure how to express it clearly but the idea is:
For an arbitrary element $y\in T(V)$, define the rank of $y$ to be the smallest number $t$ such that in all possible expression of $y$, there are elementary tensors of rank $t$.
Thus the rank of $k\oplus (v_1\otimes v_2)\oplus (v_3\otimes v_4\otimes v_5)$ is $0$; the rank of $(v_1\otimes v_2+v_1'\otimes v_2')\oplus (v_3\otimes v_4\otimes v_5)$ is $2$, etc.
Hopefully the idea is clear. How do I express this definition of rank concisely and clearly?
This is not how one would usually define the rank of an element of the tensor algebra. The tensor algebra is the direct sum $$\bigoplus_{n=0}^{\infty}{V^{\otimes n}}$$ where $V^{\otimes n}$ is $V$ tensored with itself $n$ times. The rank of an element would usually be defined as the largest integer $N$ such that if we write the element as a sum $$\sum_{n=0}^{\infty}{w_n}$$ where $w_n\in V^{\otimes n}$ for all $n$, then $w_N\neq 0$. Note that only finitely many terms in the sum can be nonzero because this is the direct sum, not the direct product. This is closely related to the concept of the degree of a polynomial, as if we take the quotient by the commutator ideal we obtain the symmetric algebra, which is isomorphic to a polynomial ring, and the symmetric tensors with $n$ factors are homogenous of degree $n$. If you wanted to go ahead and define rank to be the smallest $N$ such that $w_N$ is nonzero, you can use this answer to define it carefully.