Monomial and binomial retracts of polynomial rings

Abstract

Let $R$ be a ring and $B:=R[X_1,\dots ,X_n]$ the polynomial ring in $n$ variables over $R$. In this article, we consider retractions $\phi :B⟶B$ such that $\phi (X_i)$ is $0$, or a monomial or the sum of two monomials. We prove that, under certain conditions on the base ring $R$, the resulting retracts are also polynomial rings over $R$. We characterize different monomial retractions of $B$, i.e., where $\phi (X_i)$ is $0$ or a monomial for all $i$, giving the same retract. We also generalize the structure of monomial retracts by allowing a few variables to map to arbitrary polynomials, but only under some additional restrictions on the base ring $R$.