We provide a list of open problems that are connected to the commutative and noncommutative ring theory, K-theory and homological algebra. Some problems are solved now completely, some partially, and the most of them are without solutions, but are supplemented with ideas and hints.
The current paper contains a list of questions that were posed by the author to the participants of the scientific seminar "The Problems of Elementary Divisor Rings" over the years. The formulated questions vary from the simple checkup questions to the shorten versions of the further researches. Some questions are just the exercises, and some other are deep problems such that their solutions will be a nontrivial contribution to the general theory.
Eventually, the author would like to receive from the reader any information about the possible errors and inaccuracies in these questions. Moreover, it will be twice better to receive any ideas or even a complete solutions to the posed questions, but it is useful to remark that the answer to a question marked as "Open Problem" sometimes cannot be a sufficient reason of its publication.
One of the main sources of the almost all questions in the current paper is the rather old problem of the full description of the elementary divisor rings. The notion of the elementary divisor ring was introduced by Kaplansky in [Kaplansky I. Elementary divisors and modules // Trans. Amer. Math. Soc. - 1949. - 66. - p. 464-491]. There are a lot of researches that deal with the matrix diagonalization in the dierent cases (the most comprehensive history of these researches can be found in [Zabavsky B.V. Diagonal reduction of matrices over rings // Mathematical Studies, Monograph Series. VNTL Publishers - 2012. - volume XVI. - 251p.]). There are still a lot of publications concerning the topic. On the other hand, the most of them are outdate in the ideological aspect. This is connected with the appearance of such K-theoretical invariant as the stable range that was established in 1960s by Bass. One of the most fruitful aspects of Bass' studies was the following fact: a lot of answers to the problems of the linear algebra over the rings becomes more simple when we increase the dimension of the considered object (the rank of the projective module, the size of the matrix etc.), and, furthermore, the answer is independent from the choice of the base ring for rather big the dimensions' values, as well as is independent from the dimension of the current object - it functions only in the terms of the geometry of the given module.
Moreover, it have been discovered that in the commutative rings' case there exist structural theorems of these objects starting from some small values of stable range (for example, 1 or 2), that depends only on the considered problem, but not from the dimension or structure of the base ring.
Thus, for example, in the case of the commutative Bezout domains the elementary divisor rings are precisely the rings of neat range 1.
- Adequate properties of the elements with almost stable range 1 of a commutative elementary divisor domain
- Bezout rings of stable range 2 and square stable range 1
- A pseudo-prime elements of a commutative domain
- A pseudo-irreducible elements of a commutative domain
- Commutative Bezout domains in which any nonzero prime ideal is contained in a finite set of maximal ideals