This paper takes a new look at ideals generated by 2×2 minors of 2×3 matrices whose entriesare powers of three elements not necessarily forming a regular sequence. A special case of this is the idealsdetermining monomial curves in three-dimensional space, which were studied by Herzog. In the broadercontext studied here, these ideals are identified as Northcott ideals in the sense of Vasconcelos, and sotheir liaison properties are displayed. It is shown that they are set-theoretically complete intersections,revisiting the work of Bresinsky and of Valla. Even when the three elements are taken to be variablesin a polynomial ring in three variables over a field, this point of view gives a larger class of ideals thanjust the defining ideals of monomial curves. We then characterize when the ideals in this larger classare prime, we show that they are usually radical and, using the theory of multiplicities, we give upperbounds on the number of their minimal prime ideals, one of these primes being a uniquely determinedprime ideal of definition of a monomial curve. Finally, we provide examples of characteristic-dependentminimal prime and primary structures for these ideals.