The Zero Divisor Conjecture and Self-Injectivity for Monoid Rings
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.