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In earlier work, it was established that for any finite field k, the freeassociative k-algebra on one generator x, denoted by k[x]_0, had infinitelymany maximal T-spaces, but exactly two maximal $ideals (each of which is amaximal T-space). However, aside from these two T-ideals, no examples ofmaximal T-spaces of k[c]_0 have been identified. This paper presents, for eachfinite field k, an infinite sequence of proper T-spaces of k[x]_0 (no one ofwhich is a T-ideal), each of finite codimension, and for each one, both alinear basis for the T-space itself and a linear basis for a complementarylinear subspace are provided. Morever, it is proven that the first T-space inthe sequence is a maximal T-space of k[x]_0, thereby providing the firstexample of a maximal T-space of k[x]_0 that is not a maximal T-ideal.

Maximal T-spaces of the free associative algebra over a finite field