Department of Mathematics
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Let R be a regular ring and L a lattice. Denote by Id(R) the lattice of 2-sided ideals of R, L(R) the lattice of principal left ideals of R, Con(L) is the lattice of congruences of L and Id_c(R) the join semilattice of principal 2-sided ideals of R. I will give several talks about the following results:

Theorem: Id(R) is a distributive lattice; L(R) is a sectionally complemented modular lattice, and Id(R) is isomorphic to Con(L(R)).

Theorem: Let F be a field. If D is a distributive lattice with zero, then there is a locally matricial algebra R over F such that Id_c(R) is isomorphic (as a join semilattice with zero) to D.

Theorem: There is a distributive semilattice with zero of cardinality \Aleph_2 which is not isomorphic to Id_c(R) for any regular ring R.