Semiartinian V-Rings and Semiartinian Von Neumann Regular Rings

Abstract

Semiartinian right V-rings, which we call right SV-rings, form a special class of Von Neumann regular rings. We characterize these rings by the fact that every factor ring imbeds as a subring in a direct product of right full linear rings containing the socle. If R is a semiartinian ring with all primitive factor rings artinian, then the condition of being an SV-ring is right/left symmetrical and is equivalent to being regular. On the other hand, if R is a right and left SV-ring, then all primitive factor rings of R are artinian. For right SV-rings whose proper ideals are prime we show that the condition of being unit-regular is equivalent to being directly finite. On the other hand we show that there exists a directly finite right SV-ring which is not unit-regular. Furthermore we provide two constructions. For any given ordinal ξ, the first one gives a prime, unit-regular right SV-ring of Loewy length ξ + 1, which is not a left V-ring, and is hereditary if ξ is a natural number; the second one gives a directly infinite right SV-ring, not a left V-ring, whose Loewy length is ξ + 2 if ξ is a natural number and is ξ + 1 otherwise. These constructions are general enough to produce a wide supply of SV-rings, starting from given ones.