Some applications of a general lemma on invariant means

Zorn’s Lemma and Some Applications

Zorn’s lemma is a result in set theory that appears in proofs of some non-constructive existence theorems throughout mathematics. We will state Zorn’s lemma below and use it in later sections to prove some results in linear algebra, ring theory, and group theory. In an appendix, we will give an application to metric spaces. The statement of Zorn’s lemma is not intuitive, and some of the termino...

Hensel’s Lemma and Some Applications

The purpose of this note is to record Hensel’s Lemma and a few of its immediate applications for my Math 831 class. At the end of the 19th century Hensel introduced the notion of completion into a number theoretic context in order to bring to bear techniques of analysis on some purely algebraic problems in number theory. One of his most crucial discoveries (in modern terms) was that when a ring...

Some Applications of Schwarz Lemma for Operators

A generalized Schwarz lemma and some Harnack type inequalities for operators have been obtained in this paper.

Some Applications of Zorn’s Lemma in Algebra

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, grou...

The Cycle Lemma and Some Applications