Idealizers in free algebras

Abstract

We are concerned with the idealizer S of a principal right ideal rR in a free associative algebra R. This comes up naturally in the theory of the corresponding one-relator associative algebra R/RrR, and seems to be a reasonable route to the investigation of such algebras. In many ways, S is well-behaved. It is always a pure (in the sense of Kosevoi) subalgebra. The case where r is homogeneous seems to produce a free idealizer - we are able to demonstrate freeness in such disparate cases as r being a monomial, and r being such that R/RrR has no zero divisors. The latter result is due to Jacques and Tekla Lewin, and we will prove a more general form of it. If r is not homogeneous we know that S need not be free - indeed, not even a 2-fir. But we can give examples of cases where S is free, and one of these provides the first known example of a non-regularly embedded free subalgebra of a free algebra. For certain types of one-relator algebras we can prove analogues of Magnus' theorems on one-relator groups, and in particular we prove a result of Sirsov's. We also investigate the Golod-Safarevic power series associated to these and other algebras. With machinery we develop, we derive the non-homogeneous generalization of the Golod-Safarevic criterion for infinite dimensionality, and extend the Schreier-Lewin dimension formula to rings with weak algorithm. Finally, we append our work with Bergman extending Lewin's upper triangular matrix representations.