Representation of artinian partially ordered sets over semiartinian Von Neumann regular algebras

TL;DR

This paper explores the structure of primitive ideals in semiartinian Von Neumann regular rings, establishing a correspondence with artinian posets and constructing specific algebras with prescribed ideal and module properties.

Contribution

It introduces a construction of semiartinian, unit-regular algebras from artinian posets, linking their simple modules and ideal lattices to the poset structure.

Findings

Primitive ideals form an artinian poset with maximal chains having a greatest element.

The lattice of ideals is anti-isomorphic to the lattice of upper subsets of the primitive ideal set.

Constructed algebras have simple modules ordered by the poset, with properties depending on the poset structure.

Abstract

If $R$ is a semiartinian Von Neumann regular ring, then the set $\Prim_{R}$ of primitive ideals of $R$, ordered by inclusion, is an artinian poset in which all maximal chains have a greatest element. Moreover, if $\Prim_{R}$ has no infinite antichains, then the lattice $\BL_{2}(R)$ of all ideals of $R$ is anti-isomorphic to the lattice of all upper subsets of $\Prim_{R}$. Since the assignment $U\mapsto r_R(U)$ defines a bijection from any set $\Simp_R$ of representatives of simple right $R$-modules to $\Prim_{R}$, a natural partial order is induced in $\Simp_R$, 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 $I'\sbs I$ and a field $D$, we present a construction which produces a semiartinian and unit-regular $D$-algebra $D_I$ having the following features: (a) $\Simp_{D_I}$ is order isomorphic to $I$; (b) the assignment $H\mapsto\Simp_{D_I/H}$ realizes an anti-isomorphism from the lattice $\BL_{2}(D_I)$ to the lattice of all upper subsets of $\Simp_{D_I}$; (c) a non-maximal element of $\Simp_{D_I}$ is injective if and only if it corresponds to an element of $I'$, thus $D_I$ is a right $V$-ring if and only if $I' = I$; (d) $D_I$ is a right \emph{and} left $V$-ring if and only if $I$ is an antichain; (e) if $I$ has finite dual Krull length, then $D_I$ is (right and left) hereditary; (f) if $I$ is at most countable and $I' = \vu$, then $D_I$ is a countably dimensional $D$-algebra.