Retracts of Laurent polynomial rings

Abstract

Let $R$ be an integral domain and $B=R[x_1,\dots ,x_n]$ be the polynomial ring. In this paper, we consider retracts of $B[1/M]$ for a monomial $M$. We show that (1) if $M={\prod }_{i=1}^n{x}_i$, then every retract is a Laurent polynomial ring over $R$, (2) if $R$ is a perfect field and $n=3$, then every retract is isomorphic to $R[y_1^{\pm 1},\dots ,y_s^{\pm 1},z_1,\dots ,z_t]$ for some $s,t\ge 0$.

5 pages