Some applications of Zorn's Lemma in algebra

Abstract

We indicate some new applications of Zorn's Lemma to a number of algebraic areas. Specifically, we show that a property $P$ holds for all the subobjects of a given object if and only if $P$ supports both the chain condition from Zorn's Lemma and some finitistic conditions on subobjects that have the flavor of mathematical induction. The specific algebraic contexts considered are modules, sets, groups, algebras, and integral domains. Particular emphasis is given to applications involving star operations on integral domains.