The study of ideals of rings is a classical branch of the commutative algebra, which began under the influence of the theory of numbers and algebraic geometry, and in the 20th century, in works of E. Noeter, E. Artin, A. Speiser and others, formed an independent area.
For the integrally closed (non-special) rings, this theory has gained completeness, whereas for rings with singularities there are many unsolved problems. In particular, this is the question of the number and structure of classes of ideals. The first results here were obtained by D. Faddeev and Z. Borevich for quadratic rings, and by them and H. Bass for a wide class of rings, now known as bassian. It was found that for any such rings any ideal is invertible (over its ring of multipliers). Later, D. Faddeev showed that for each cubic ring, each ideal is either invertible or dual to invertible, and Yu. Drozd summed this result for a wide class of rings. G. Jakobinsky, Yu. Drozd and A. Roiter gave the criterion that the ring has a finite number of classes of ideals. In the work of G.-M. Greuel and G. Knoerrer the relations with the classification of singularities by V. Arnold were established, in particular, it was shown that a singularities with a finite number of classes of ideals are those which dominate simple singularities in the sense of Arnold. For singularities with only one-parameter families of ideals Schappert, Drozd and Greuel showed that they are those which dominate unimodal or bimodal singularities. Further research in this direction, to which is devoted the dissertation, is promising and important both for the theory of ideals itself and for the theory of singularities and related branches of algebraic geometry.
The main results of the dissertation are related to the classification of cubic rings and singularities with two-parameter families of ideals. The first branch gives an overview of the theory of ideals and provides the famous technical results used in the work. In the second branch, cubic rings are considered, that is, the extensions of a local dedekind ring contained in a cubic expansion of its field of fractions. It is here that a complete description of local cubic rings is given, as well as of their ideals. In particular, in the geometric case, that is for local rings of algebraic curves over an algebraically closed field, the maximum number of parameters in the families of ideals is calculated.
