Uniqueness of generators of principal ideals in rings of continuous functions

Let $aR$ denote the principal right ideal generated in a ring $R$ by an element $a$. Kaplansky has raised the question: If $aR = bR$, are $a$ and $b$ necessarily right associates? In this note we show that for rings of continuous functions the answer is affirmative if and only if the underlying topological space is zero-dimensional. This gives an algebraic characterization of the topological concept “zero-dimensional". By extending the notion of uniqueness of generators of principal ideals we are able to give an algebraic characterization of the concept “$n$-dimensional".