Zorn's lemma and algebraic structures

Zorn’s lemma is a principle in mathematics used to prove the existence of certain objects when direct construction is difficult. It states that if every chain (or collection) of objects has an upper bound, then there must be at least one maximal object in the collection—one that cannot be extended further. This helps mathematicians establish the existence of things like bases in vector spaces or maximal ideals in rings. Algebraic structures, such as groups, rings, or fields, are systems with elements and operations satisfying specific rules, forming the foundation for much of modern algebra.