Insa Cremer – Strong semilattice decomposition of m-domain rings

University of Latvia

A semigroup is said to be a D-semigroup if there exists
some subset of its set of idempotents such that, for every , there exists the smallest in the sense of the usual partial order of idempotents with the property that . So on every D-semigroup, we can define a unary operation assigning to each the smallest idempotent with .
An m-domain ring is a ring such that, for every element , there exists a central idempotent such that , and, if is an idempotent such that , then . The multiplicative semigroup of an m-domain ring is a D-semigroup (with the unary operation ).

Given an m-domain ring , we can decompose it into an inverse system of pairwise disjoint right-cancellative D-semigroups with identities over the set of idempotent elements of . In the talk, the following results on this decomposition will be presented.

First, the strong semilattice of D-semigroups obtained from this inverse system is the multiplicative D-semigroup of the ring .

Second, let be an inverse system of pairwise disjoint right-cancellative D-semigroups with identities over some lower semilattice . If the strong semilattice of D-semigroups obtained from the inverse system is the multiplicative D-semigroup of some m-domain ring , then the inverse system is the decomosition of the m-domain ring .