Let $A$ be an unital associative algebra generated by some elements $a_1,...,a_n$. Is it always possible to come up with a $\mathbb{Z}_2$-grading for $A$?
For example does it work if I consider the $a_i$ to be parity odd (and extend it in the sense that $a_ia_j$ is even etc.) and the unit to be parity even?
Consider the algebra $A = \mathbb{R}[x] / (x^2-x)$. In this algebra, $x^2=x$. How would you assign a parity to $x$ in this case?