6 - Centralizers and subalgebras

Summary

The first topic of this chapter is commutativity in firs. We shall find that any maximal commutative subring of a 2-fir with strong DFL is integrally closed (Corollary 1.2), and the same method allows us to describe the centres of 2-firs as integrally closed rings and make a study of invariant elements in 2-firs and their factors in Sections 6.1 and 6.2. The well-known result that a simple proper homomorphic image of a principal ideal domain is a matrix ring over a skew field is generalized here to atomic 2-firs (Theorem 2.4). In Section 6.3 the centres of principal ideal domains are characterized as Krull domains. Further, the centre of a non-principal fir is shown to be a field in Section 6.4.

Secondly we look at subalgebras and ideals of free algebras in Section 6.6; by way of preparation submonoids of free monoids are treated in section 6.5. A brief excursion into coding theory shows how the Kraft–McMillan inequality can be used to find free subalgebras, and the fir property of free algebras is again derived (Theorem 6.7). Section 6.7 is devoted to a fundamental theorem on free algebras: Bergman's centralizer theorem (Theorem 7.7).

Section 6.8 deals with invariants under automorphisms of free algebras, and Section 6.9 treats the Galois correspondence between automorphism groups and free subalgebras, as described by Kharchenko.