Hua´s identity and Gröbner bases

Abstract

In einem assoziativen Ring mit 1 gilt für a, b Elemente des Ringes, dass wenn 1-ab ein Rechtsinverses x besitzt, so ist y=1+axb rechtsinvers von 1+ba. Um die Frage systematisch zu beantworten, kann man in einem Ring Z[a,b,x]/I arbeiten wo a,b,x nicht kommutierende Unbekannte sind, und I ein zweiseitiges durch (1-ab)x-1 erzeugtes Ideal ist. Man sucht dann y, sodass (1-ba)y-1 in I liegt. Dazu kann man Gröbnerbasentechniken anwenden.<br />Die selbe Technik kann auf andere etwas kompliziertere Aufgaben angewendet werden wie auf Hua´s Identität.<br />

It is a well-known fact that in any associative ring with unity 1-ab with a,b elements in the ring having a right inverse x allows us to immediately verify that y=1+axb is a right inverse for 1+ba. If one "does not know" the solution y one could ask whether or not there is a systematic way to find it. To this end we employ the following idea: We compute in the ring Z[a,b,x]/I where a,b,x are non-commuting indeterminates and I is the two-sides ideal generated by (1+ab)x-1. Then we are interested to find y with (1+ba)y-1belongs to I. Having found y the ring-homomorphism that sends a,b and x to the respective elements in an arbitrary associative ring so that (1+ab)x=1 yields the homomorphic image to be a solution - establishing the fact.<br /> In order to "find" y we employ a Gröbner basis technique. As known, any such ideal basis provides an effective tool for deciding membership of an element in an ideal.<br /> As a second slightly more complicated example of how to employ such Gröbner basis technique we derive Hua´s identity. This identity states that (a+a(b -1)a) -1+(a+b) -1=a -1 provided a,b and a+b have inverses.<br />Again, this identity can be verified in an elementary manner. On the other hand, if it is known that a,b and a+b have inverses it is far from obvious how to find an inverse of a+a(b -1)a.<br />