Zornova lema in njena uporaba - Zupančič, Žiga (Matrika, 2018)
In this article Zorn's lemma and some equivalent formulations are presented. An application in group theory is presented by proving that homomorphisms from subgroups of Abelian groups to divisible groups can be extended to the whole Abelian group. In ring theory Zorn's lemma implies the existence of maximal left ideals in rings. Also, it implies that the intersection of all prime ideals in a commutative ring is equal to the set of nilpotent elements of the ring. Cohen's theorem and it's generalization are also proved. Using Zorn's lemma, it is shown, that every nonzero module over a division ring has a basis. The consequence of that is that every vector space over a field has a basis. This corollary is used to prove that a group with more than two elements has a nontrivial automorphism. The use of Zorn's lemma in graph theory is shown when proving that every connected graph contains a spanning tree.
Keywords: aksiom izbire; axiom of choice; delna urejenost; linearna urejenost; maksimalni element; matematika; mathematics; maximal element; partial ordering; total ordering; Zorn lemma; Zornova lema