  This paper takes a new look at ideals generated by 2x2 minors of 2x3 matrices
whose entries are powers of three elements not necessarily forming a regular
sequence. A special case of this are the ideals determining monomial curves in
three dimensional space, which were already studied by Herzog. In the broader
context studied here, these ideals are identified as Northcott ideals in the
sense of Vasconcelos, and so their 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
variables in a polynomial ring in three variables over a field, this point of
view gives a larger class of ideals than just the defining ideals of monomial
curves. We then characterize when the ideals in this larger class are prime, we
show that they are usually radical and, using the theory of multiplicities, we
give upper bounds on the number of their minimal prime ideals, one of these
primes being a uniquely determined prime ideal of definition of a monomial
curve. Finally, we provide examples of characteristic-dependent minimal prime
and primary structures for these ideals.
