Abstract:
This paper deals with one-dimensional (commutative) rings without nilpotent elements such that every ideal is generated by three elements. It is shown that in such rings the square of every ideal is invertible, i.e. divides its multiplier ring. In addition, every ideal is distinguished, in the sense that on localization at any maximal ideal it becomes either principal or dual to a principal ideal. Conversely, if a one-dimensional ring without nilpotent elements satisfies either of these conditions, and if all its residue class fields are $2$-perfect and contain at least three elements, then every ideal can be generated by three elements.
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