Bilinear maps on the ring of strictly upper triangular matrices

TL;DR

This paper characterizes bilinear maps on strictly upper triangular matrices over a ring that satisfy a specific commutation identity, extending previous linear map results to bilinear maps.

Contribution

It provides a complete description of bilinear maps on upper triangular matrices satisfying a commutation condition, generalizing known linear map results.

Findings

Characterization of bilinear maps satisfying the identity $[f(X,X),X]=0$.

Extension of linear map results to bilinear maps.

Complete description of such bilinear maps on $N_n(R)$.

Abstract

Let $R$ be a 2-torsion free unital ring and $N_n=N_n(R)$ the ring of strictly upper triangular matrices with entries in $R$ and center $Z=Z(N_n)$. It has been previously shown that any linear map $f:N_n\rightarrow N_n$ satisfying the condition $[f(X),X]=0$ must be of the form $f(X)=\lambda X+\mu(X)$ for some $\lambda\in R$ and additive map $\mu$ defined on $N_n$. We extend these known results by providing a complete description of the bilinear maps $f:N_n\times N_n\rightarrow N_n$ satisfying the identity $[f(X,X),X]=0$ for all $X\in N_n$.