Forms of Choice in Ring Theory

We investigate the relationship between various choice principles and nth -root functions in rings. For example, we show that the Axiom of Choice is equivalent to the statement that every ring has a square-root function. Furthermore, we introduce a choice principle which implies that every integral domain has an nth -root function (for odd integers n), and introduce another choice principle which is equivalent to the Prime Ideal Theorem restricted to certain ideals. Finally, we investigate the dependencies between the two new choice principles and a choice principle for families of n-element sets.