The Zero Divisor Conjecture and Self-Injectivity for Monoid Rings

Joe Sullivan

Let G be a monoid and let k be a field. The monoid ring kG is studied. The well known zero divisor conjecture, still open, is that if G is a torsion free group, then kG is a domain. It is shown that this is false if G is only a monoid. It is proved that if G is a non-cancellative monoid, then kG is not a domain. On the other hand it is known that if G is cancellative and abelian, then G can be embedded in a group, and it follows that if G is torsion-free, then kG is a domain.

Next nilpotent elements, units and semisimplicity of a torsion-free monoid are studied. Finally it is well known that if G is a finite group, then kG is self-injective. In the case of finite monoids, examples are presented for which kG is not self-injective, and the injective hull of kG is obtained. Finally hereditary properties of kG are studied.