Semiartinian Von Neumann regular rings with prescribed poset of representatives of simple modules.
by
Giuseppe Baccella
Dipartimento di Matematica Pura ed Applicata. Università di L'Aquila. 67100 L'AQUILA - ITALY

Let R be a semiartinian Von Neumann regular ring and let Simp_R denote a set of representatives of simple right R-modules. Then Simp_R has a natural structure of poset which influences the order structure of the Grothendieck group of R. The maximal elements of Simp_R are precisely those simple modules which are finite dimensional as vector spaces over their endomorphism division rings. If S is a ring Morita equivalent to R, then Simp_S and Simp_R are isomorphic as posets. By writing I = Simp_R, then it turns out that: (a) I is artinian, (b) every maximal chain of I has a maximal element, (c) the dual classical Krull dimension \xi of I is a successor ordinal and I_(\xi)\I_(\xi-1) is finite (here I_(\alpha) denotes the \alpha-th term of the dual classical Krull filtration of I).

Conversely, let I be a poset satisfying (a), (b), (c). Given a field F, there exist two unit-regular and semiartinian F-algebras R and S such that Simp_R and Simp_S are order isomorphic to I; moreover R is a right V-ring, while the only simple injective right S-modules are the maximal elements of Simp_S. If I is finite or is countable, then S is countably dimensional over F.

Date received: March 23, 1999