Let $K$ be a field.
I want to express the ring that contains only Laurent polynomials $$P(X)=\displaystyle\sum_{i\geq 0} p_iX^i + \displaystyle\sum_{j> 0}p_{-j}X^{-j}\in K[X^{\pm 1}]$$ with the constraint $p_i\neq -p_i \forall i\in \mathbb{Z}_{\geq 1}.$
Put another way, I would like the ring of Laurent polynomials modulo the polynomials $g(X)$, such that $g(X)=-g(X^{-1})$.
Can we write the desired ring as a quotient of $K[X^{\pm 1}]$?