## 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.