Semiartinian \(V\)-rings and semiartinian von Neumann regular rings
A ring \(R\) is called a right \(SV\)-ring if \(R\) is a right semiartinian ring (i.e. every nonzero right module has nonzero socle) and a right \(V\)- ring (i.e. every simple right module is injective). The paper under review presents an extensive investigation on the class of right \(SV\)- rings, which in fact form a special class of von Neumann regular rings. The author proves that if \(R\) is a semiartinian ring with all primitive factor rings artinian, then the condition of being an \(SV\)-ring is left/right symmetrical, and is equivalent to being von Neumann regular. He also proves that if \(R\) is a strongly primitive right \(SV\)-ring, then \(R\) is directly finite if and only if \(R\) is unit-regular. Several examples are given which illustrate the concepts considered in the paper. In particular, the author provides two constructions which produce a wide supply of right \(SV\)-rings. For any ordinal \(\xi\), the first one shows that there exists a prime, unit-regular, right \(SV\)-ring of Loewy length \(\xi+1\), which is not a left \(V\)-ring, and is hereditary if \(\xi\) is a natural number. The second one gives a directly infinite right \(SV\)-ring, not a left \(V\)-ring, whose Loewy length is \(\xi+2\) if \(\xi\) is a natural number and is \(\xi+1\) otherwise.
scientific article; zbMATH DE number 763962