Monomial retracts of polynomial rings are polynomial rings

TL;DR

This work resolves when monomial retracts of $R[X_1,\dots,X_n]$ are themselves polynomial rings by tying monic monomial retractions to idempotent exponent matrices and leveraging a canonical standard form. Under mild conditions on $R$ (no nontrivial idempotents; in particular, when $R$ is an integral domain), every nondegenerate monomial retract has image isomorphic to a polynomial ring $R^{[p]}$ for some $p\le n$, with a constructive isomorphism to $R^{[p]}$. The paper also provides a complete classification of when different monic monomial retractions yield the same retract, including a combinatorial count via sets $\Gamma_j$, which sharpens understanding of the retract structure and its implications for related cancellation questions.

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 $\varphi : B \longrightarrow B$ such that $\varphi(X_i)$ is either a monic monomial or $0$. We prove that if $R$ is an integral domain, then any such retract is isomorphic to $R^{[p]}$, the polynomial ring in $p$ variables over $R$, for some $0 \le p \le n$. We also characterize different monomial retractions of $B$ which give the same retract.

Theorems & Definitions (14)

• Lemma 2.1
• proof
• Proposition 2.2
• Theorem 3.1
• proof
• Remark 3.2
• Corollary 3.3
• proof
• Proposition 4.1
• proof