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

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

2011-04-25

arxiv.org/abs/1104.4755v1

Mathematics

Rings and Algebras

Scientific paper

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

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