Representation of artinian partially ordered sets over semiartinian von Neuman regular algebras

If R is a semiartinian von Neumann regular ring, then the set PrimR of primitive ideals of R, ordered by inclusion, is an artinian poset in which all maximal chains have a greatest element. Moreover, if PrimR has no infinite antichains, then the lattice L2(R) of all ideals of R is anti-isomorphic to the lattice of all upper subsets of PrimR. Since the assignment U↦rR(U) defines a bijection from any set SimpR of representatives of simple right R-modules to PrimR, a natural partial order is induced in SimpR, under which the maximal elements are precisely those simple right R-modules which are finite dimensional over the respective endomorphism division rings; these are always R-injective. Given any artinian poset I with at least two elements and having a finite cofinal subset, a lower subset I′⊂I and a field D, we present a construction which produces a semiartinian and unit-regular D-algebra DI having the following features: (a) SimpDI is order isomorphic to I; (b) the assignment H↦SimpDI/H realizes an anti-isomorphism from the lattice L2(DI) to the lattice of all upper subsets of SimpDI; (c) a non-maximal element of SimpDI is injective if and only if it corresponds to an element of I′, thus DI is a right V-ring if and only if I′=I; (d) DI is a right and left V-ring if and only if I is an antichain; (e) if I has finite dual Krull length, then DI is (right and left) hereditary; (f) if I is at most countable and I′=∅, then DI is a countably dimensional D-algebra.