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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 whichall maximal chains have a greatest element. Moreover, if $\Prim_{R}$ has noinfinite antichains, then the lattice $\BL_{2}(R)$ of all ideals of $R$ isanti-isomorphic to the lattice of all upper subsets of $\Prim_{R}$. Since theassignment $U\mapsto r_R(U)$ defines a bijection from any set $\Simp_R$ ofrepresentatives of simple right $R$-modules to $\Prim_{R}$, a natural partialorder is induced in $\Simp_R$, under which the maximal elements are preciselythose simple right $R$-modules which are finite dimensional over the respectiveendomorphism division rings; these are always $R$-injective. Given any artinianposet $I$ with at least two elements and having a finite cofinal subset, alower subset $I'\sbs I$ and a field $D$, we present a construction whichproduces a semiartinian and unit-regular $D$-algebra $D_I$ having the followingfeatures: (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) anon-maximal element of $\Simp_{D_I}$ is injective if and only if it correspondsto 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 anantichain; (e) if $I$ has finite dual Krull length, then $D_I$ is (right andleft) hereditary; (f) if $I$ is at most countable and $I' = \vu$, then $D_I$ isa countably dimensional $D$-algebra.

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