For any algebraic theory that is expressible in first order logic in a countable language, and this includes groups, rings, fields, partial orders, lattices, etc. etc., then the basic fact is expressed by the Lowenheim-Skolem theorem, which asserts that if the theory has an infinite model, then it has models of every infinite cardinality. In general, one gets models of the theory of every cardinality above the size of the language (and this covers your $R$-module case).
One needs the Axiom of Choice to prove this, however, and this use is necessary, since the Axiom of Choice is equivalent to the assertion that every set carries a group structure, as explained in the answer to this MO question.