Unions of Chains of Primes
Be'eri Greenfeld, Louis H. Rowen, Uzi Vishne
TL;DR
This paper investigates the conditions under which unions of chains of prime ideals remain prime, exploring independence from semiprimes and establishing bounds related to PI-algebras.
Contribution
It demonstrates the independence of prime union properties from semiprime properties and establishes a PI-class bound on non-prime unions in PI-algebras.
Findings
Union of prime chains is not always prime.
Prime union property is independent of semiprime union property.
PI-class bounds the number of non-prime unions in PI-algebras.
Abstract
The union of an ascending chain of prime ideals is not always prime. We show that this property is independent of the parallel property for semiprimes. We also show that the PI-class is a tight bound on the number of non-prime unions of subchains in a chain of primes in a PI-algebra.
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