### A maximal \(T\)-space of \(\mathbb{F}_{3}[x]_0\)

#### Abstract

In earlier work, we have established that for any finite field \(k\), the free associative \(k\)-algebra on one generator \(x\), denoted by \(k[x]_0\), has infinitely many maximal \(T\)-spaces, but exactly two maximal \(T\)-ideals (each of which is a maximal \(T\)-space). However, aside from these two \(T\)-ideals, no specific examples of maximal \(T\)-spaces of \(k[x]_0\) were determined at that time. In a subsequent work, we proposed that for a finite field \(k\) of characteristic \(p>2\) and order \(q\), for each positive integer \(n\) which is a power of 2, the \(T\)-space \(W_n\), generated by \(\{x+x^{q^n}, x^{q^n+1}\}\), is maximal, and we proved that \(W_1\) is maximal. In this note, we prove that for \(q=p=3\), \(W_2\) is maximal.

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