Kaplansky made the
following conjecture: A ring R is von Neumann regular if and only if R is
semiprime and each prime factor ring of R is von Neumann regular. That
the conjecture failed in general was shown by a counterexample of Snider.
It is established that a necessary and sufficient condition for Kaplansky’
conjecture to hold is that the union of every chain of semiprime ideals of R
be semiprime. From this theorem flow many corollaries including proofs of
Kaplansky’s conjecture in the known special cases of commutative rings and rings
without nilpotent elements. Moreover, it is used repeatedly in previously
unaccessable situations. In one such situation semiprime Azumaya algebras with
prime ideals maximal are characterized as von Neumann regular algebras. In
another a theorem of Armendariz and Fisher is proved which characterizes von
Neumann regular polynomial identity rings (P. I.-rings) as fully idempotent
rings.
