Related papers: On minimal extensions of rings and applications
We develop a theory of extensions of hyperfields that generalizes the notion of field extensions. Since hyperfields have a multivalued addition, we must consider two kinds of extensions that we call weak hyperfield extensions and strong…
We raise some questions about graph polynomials, highlighting concepts and phenomena that may merit consideration in the development of a general theory. Our questions are mainly of three types: When do graph polynomials have reduction…
In what follows we generalize the notion of a complemented ring to rings that are not necessarily reduced. We then determine how our concepts fit in with other well-known classes of rings.
We prove extension-dimensional versions of finite dimensional selection and approximation theorems. As applications, we obtain several results on extension dimension.
We give an algorithm that constructs a minimal set of polynomials defining all extension of a $(\pi)$-adic field with given, inertia degree, ramification index, discriminant, ramification polygon, and residual polynomials of the segments of…
The use of nonstandard methods to characterize properties of weak, strong and mixed extensions of congruences to ultrafilters has been the main topic of several recent papers. We show that similar methods can be used to characterize the…
We introduce a new method of constructing complete sequences of key polynomials for simple extensions of tame fields. In our approach the key polynomials are taken to be the minimal polynomials over the base field of suitably constructed…
In a paper on the taxonomy of 2-primal rings, examples of various types of rings that are related to commutativity such as reduced, symmetric, duo, reversible and PS~I were given in order to show that the ring class inclusions were strict.…
Among Thurston maps (orientation-preserving, postcritically finite branched coverings of the 2-sphere to itself), those that arise as subdivision maps of a finite subdivision rule form a special family. For such maps, we investigate…
The notion of trivial extension of a ring by a module has been extensively studied and used in ring theory as well as in various other areas of research like cohomology theory, representation theory, category theory and homological algebra.…
In this paper, we introduce a strong property $(A)$ and we study the transfer of property $(A)$ and strong property $(A)$ in trivial ring extensions and amalgamated duplication of a ring along an ideal. We also exhibit a class of rings…
We obtain necessary and sufficient conditions for abelian varieties to acquire semistable reduction over fields of low degree. Our criteria are expressed in terms of torsion points of small order defined over unramified extensions.
We extend the notion of type sequence to rings that are not necessarily residually rational. Using this invariant we characterize different types of rings as almost Gorenstein rings and rings of maximal length.
We derive inclusion regions for the eigenvalues of matrix polynomials expressed in a general polynomial basis, which can lead to significantly better results than traditional bounds. We present several applications to engineering problems.
Some basic properties of the ring of integers $\mathbb{Z}$ are extended to entire rings. In particular, arithmetic in entire principal rings is very similar than arithmetic in the ring of integers $\mathbb{Z}$. These arithmetic properties…
The set of minimal primes of a ring is a very important set as far the spectrum of a ring is concerned as every prime contains a minimal prime. So, knowing the minimal primes is the first (important and difficult) step in describing the…
We obtain tight bounds for the minimal number of generators of an ideal with bounded-degree generators in a polynomial ring $K[X_1,\dots,X_n],$ as well as a sharp quantification of the maximum possible size of a minimal generating set of…
We study d-minimal expansions of ordered fields, and dense pairs thereof. We also consider other generalizations of o-minimality.
The main result of this paper is that in order to prove the local uniformization theorem for local rings it is enough to prove it for rank one valuations. Our proof does not depend on the nature of the class of local rings for which we want…
The covering number of an associative ring $R$ is the minimal number of proper subrings whose union is $R$. We establish a strategy to classify unital rings of a given finite covering number, and obtain a classification of unital rings…