Vol. 54, No. 1, 1974

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On the von Neumann regularity of rings with regular prime factor rings

Joe Wayne Fisher and Robert L. Snider

Vol. 54 (1974), No. 1, 135–144

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.

Mathematical Subject Classification
Primary: 16A30
Received: 12 March 1973
Revised: 2 October 1973
Published: 1 September 1974
Joe Wayne Fisher
Robert L. Snider