Cardinality of groups and rings via the idempotency of infinite cardinals

TL;DR

This paper explores the implications of the idempotency of infinite cardinals in ZFC, deriving algebraic results about groups and rings, including cardinalities, properties of balanced rings, and classifications of certain ring types.

Contribution

It introduces the concept of balanced rings, characterizes when rings are balanced, and computes cardinalities of various algebraic structures using set-theoretic principles.

Findings

Infinite Abelian groups have proper subgroups of the same cardinality unless they are Prüfer groups.

Cardinality of monoid rings and polynomial rings with any number of variables is explicitly computed.

Zero-dimensional rings and certain classes of rings are shown to be balanced rings.

Abstract

An important classical result in ZFC asserts that every infinite cardinal number is idempotent. Using this fact, we obtain several algebraic results in this article. The first result asserts that an infinite Abelian group has a proper subgroup with the same cardinality if and only if it is not a Pr\"ufer group. In the second result, the cardinality of any monoid-ring $R[M]$ (not necessarily commutative) is calculated. In particular, the cardinality of every polynomial ring with any number of variables (possibly infinite) is easily computed. Next, it is shown that every commutative ring and its total ring of fractions have the same cardinality. This set-theoretic observation leads us to a notion in ring theory that we call a balanced ring (i.e. a ring that is canonically isomorphic to its total ring of fractions). Every zero-dimensional ring is a balanced ring. Then we show that a Noetherian ring is a balanced ring if and only if its localization at every maximal ideal has zero depth. It is also proved that every self-injective ring (injective as a module over itself) is a balanced ring.