Principal Ideal Domain Research Articles

A Note on Finitely Generated Modules over a <i>PID</i>

In this note, we provide an effective proof of the fundamental structure theorem of finitely generated modules over a principal ideal domain, from which we find the minimality of decomposition for a finitely generated module over a principal ideal domain.

On equivalence and factorization of the Kronecker product of matrices

The question of the equivalence of Kronecker products of matrices over the commutative principal ideal domain and the factorization of the Kronecker product of polynomial matrices are investigated. Cite as: V. R. Zelisko, H. V. Zelisko, On equivalence and factorization of the Kronecker product of matrices, Prykl. Probl. Mekh. Mat. , Issue 17, 11–16 (2019) (in Ukrainian), https://doi.org/10.15407/apmm2019.17.11-16

A Signature-Based Algorithm for Computing Gr\xf6bner Bases over Principal Ideal Domains

Signature-based algorithms have become a standard approach for Gröbner basis computations for polynomial systems over fields, but how to extend these techniques to coefficients in general rings is not yet as well understood. In this paper, we present a proof-of-concept signature-based algorithm for computing Gröbner bases over commutative integral domains. It is adapted from a general version of Möller’s algorithm (J Symb Comput 6(2–3), 345–359, 1988) which considers reductions by multiple polynomials at each step. This algorithm performs reductions with non-decreasing signatures, and in particular, signature drops do not occur. When the coefficients are from a principal ideal domain (e.g. the ring of integers or the ring of univariate polynomials over a field), we prove correctness and termination of the algorithm, and we show how to use signature properties to implement classic signature-based criteria to eliminate some redundant reductions. In particular, if the input is a regular sequence, the algorithm operates without any reduction to 0. We have written a toy implementation of the algorithm in Magma. Early experimental results suggest that the algorithm might even be correct and terminate in a more general setting, for polynomials over a unique factorization domain (e.g. the ring of multivariate polynomials over a field or a PID).

Efficient Gröbner bases computation over principal ideal rings

In this paper we present new techniques for improving the computation of strong Gröbner bases over a principal ideal ring R. More precisely, we describe how to lift a strong Gröbner basis along a canonical projection R→R/n, n≠0, and along a ring isomorphism R→R1×R2. We then apply this to the computation of strong Gröbner bases over a non-trivial quotient of a principal ideal domain R/nR. The idea is to run a standard Gröbner basis algorithm pretending R/nR to be field. If we discover a non-invertible leading coefficient c, we use this information to try to split n=ab with coprime a,b. If this is possible, we recursively reduce the original computation to two strong Gröbner bases computations over R/aR and R/bR respectively. If no such c is discovered, the returned Gröbner basis is already a strong Gröbner basis for the input ideal over R/nR.

On the Divisibility of Matrices with Remainder over the Domain of Principal Ideals

We study the problem of divisibility of matrices with remainder over a domain of principal ideals R and establish the conditions under which, for a pair of (n × n)-matrices A and B over the domain R , there exists a unique pair of (n × n)-matrices P and Q over R such that B = AP +Q. The application of the obtained results to finding special solutions of a Sylvester-type matrix equation is presented.

Two Notes on Automorphisms of Modules over a PID

Let D be a principal ideal domain (PID) and M be a module over D. We prove the following two dual results: (i) If M is finitely generated and x, y are two elements in M such that [Formula: see text], then there exists an automorphism α of M such that [Formula: see text]. (ii) If M satisfies the minimal condition on submodules and X, Y are two locally cyclic submodules of M such that [Formula: see text] and [Formula: see text], then there exists an automorphism α of M such that [Formula: see text].

Dedekind’s Criterion and Integral Bases

Abstract Let R be a principal ideal domain with quotient field K, and L = K(α), where α is a root of a monic irreducible polynomial F (x) ∈ R[x]. Let ℤ L be the integral closure of R in L. In this paper, for every prime p of R, we give a new efficient version of Dedekind’s criterion in R, i.e., necessary and sufficient conditions on F (x) to have p not dividing the index [ℤ L : R[α]], for every prime p of R. Some computational examples are given for R = ℤ.

A question about higher order z-ideals in commutative rings

Let D be a principal ideal domain, p an irreducible element of D and m a nonzero positive integer. In this paper we show that the ring D/pm D answers the question asked by Dube and Ighedo in [1] concerning higher order z-ideals.

Half-exact coherent functors over Dedekind domains

Let [Formula: see text] be a principal ideal domain (PID) or more generally a Dedekind domain and let [Formula: see text] be a coherent functor from the category of finitely generated [Formula: see text]-modules to itself. We classify the half-exact coherent functors [Formula: see text]. In particular, we show that if [Formula: see text] is a half-exact coherent functor over a Dedekind domain [Formula: see text], then [Formula: see text] is a direct sum of functors of the form [Formula: see text], [Formula: see text] and [Formula: see text], where [Formula: see text] is a finitely generated projective [Formula: see text]-module, [Formula: see text] a nonzero prime ideal in [Formula: see text] and [Formula: see text].

Symmetric Monoidal Structure with Local Character is a Property

In previous work we proved that, for categories of free finite-dimensional modules over a commutative semiring, linear compact-closed symmetric monoidal structure is a property, rather than a structure. That is, if there is such a structure, then it is uniquely defined (up to monoidal equivalence). Here we provide a novel unifying category-theoretic notion of symmetric monoidal structure with local character, which we prove to be a property for a much broader spectrum of categorical examples, including the infinite-dimensional case of relations over a quantale and the non-free case of finitely generated modules over a principal ideal domain.

An elementary proof of a principal ideal domain which is not an Euclidean domain

An elementary proof of a principal ideal domain which is not an Euclidean domain

Lattices Over Algebraic Integers With an Application to Compute-and-Forward

In this paper, we extend Construction A of lattices to the ring of algebraic integers of a general imaginary quadratic field that may not form a principal ideal domain (PID). We show that such a construction can produce good lattices for coding in the sense of Poltyrev and for MSE quantization. As an application, we then apply the proposed lattices to the compute-and-forward paradigm with limited feedback. Without feedback, compute-and-forward is typically realized with lattice codes over the ring of integers, the ring of Gaussian integers, or the ring of Eisenstein integers, which are all PIDs. A novel scheme called adaptive compute-and-forward is proposed to exploit the limited feedback about the channel state by working with the best ring of imaginary quadratic integers. Simulation results show that by adaptively choosing the best ring among the considered ones according to the limited feedback, the proposed adaptive compute-and-forward provides a better performance than that provided by the conventional compute-and-forward scheme which works over Gaussian or Eisenstein integers solely.

Non-existence of Some Nearly Perfect Sequences, Near Butson–Hadamard Matrices, and Near Conference Matrices

In this paper we study the non-existence problem of (nearly) perfect (almost) m-ary sequences via their connection to (near) Butson–Hadamard (BH) matrices and (near) conference matrices. We refine the idea of Brock on the unsolvability of certain equations in the case of cyclotomic number fields whose ring of integers is not a principal ideal domain and get many new non-existence results for near BH matrices and near conference matrices. We also apply previous results on vanishing sums of roots of unity and self conjugacy condition to derive non-existence results for near BH matrices and near conference matrices.

Warning’s second theorem with relaxed outputs

We present a generalization of Warning’s second theorem to polynomial systems over a finite local principal ring with restricted input and relaxed output variables. This generalizes a recent result with Forrow and Schmitt (and gives a new proof of that result). Applications to additive group theory, graph theory and polynomial interpolation are pursued in detail.

On the structure of least common multiple matrices from some class of matrices

For non-singular matrices with some restrictions, we establish the relationships between Smith normal forms and transforming matrices (a invertible matrices that transform the matrix to its Smith normal form) of two matrices with corresponding matrices of their least common right multiple over a commutative principal ideal domains. Thus, for such a class of matrices, given answer to the well-known task of M. Newman. Moreover, for such matrices, received a new method for finding their least common right multiple which is based on the search for its Smith normal form and transforming matrices.

Vector Bundles on P1ℤ with Generic Fiber O ⊕ O(1) and Simple Jumps

Vector bundles on the projective line over a Dedekind domain A are studied. In the case where A is a principal ideal domain, a complete classification is obtained for rank 2 vector bundles with generic fiber O ⊕ O(1) and special fibers isomorphic either to O ⊕ O(1) or to O(−1) ⊕ O(2).

Nice derivations over principal ideal domains

In this paper we investigate to what extent the results of Z. Wang and D. Daigle on “nice derivations” of the polynomial ring k[X,Y,Z] over a field k of characteristic zero extend to the polynomial ring R[X,Y,Z] over a PID R, containing the field of rational numbers. One of our results shows that the kernel of a nice derivation on k[X1,X2,X3,X4] of rank at most three is a polynomial ring over k.

Products of elementary matrices and non-Euclidean principal ideal domains

A classical problem, originated by Cohn's 1966 paper [1], is to characterize the integral domains R satisfying the property: (GEn) “every invertible n×n matrix with entries in R is a product of elementary matrices”. Cohn called these rings generalized Euclidean, since the classical Euclidean rings do satisfy (GEn) for every n>0. Important results on algebraic number fields motivated a natural conjecture: a non-Euclidean principal ideal domain R does not satisfy (GEn) for some n>0. We verify this conjecture for two important classes of non-Euclidean principal ideal domains: (1) the coordinate rings of special algebraic curves, among them the elliptic curves having only one rational point; (2) the non-Euclidean PID's constructed by a fixed procedure, described in Anderson's 1988 paper [2].

Isonoetherian power series rings

ABSTRACTFacchini and Nazemian proved that a valuation domain is isonoetherian if and only if it is discrete of Krull dimension ≤2 and they showed that this cannot be generalized from the local case to the global case: the 2-dimensional generalized Dedekind domain ℤ+Xℚ[[X]] is not isonoetherian. Let D be an integral domain with quotient field K. We provide necessary and sufficient conditions on D and K, so that the ring D+XK[[X]] is isonoetherian. We deduce that if D is integrally closed, then D+XK[[X]] is isonoetherian if and only if D is a semi-local principal ideal domain.

Virtually semisimple modules and a generalization of the Wedderburn-Artin theorem

ABSTRACTA widely used result of Wedderburn and Artin states that “every left ideal of a ring R is a direct summand of R if and only if R has a unique decomposition as a finite direct product of matrix rings over division rings.” Motivated by this, we call a module M virtually semisimple if every submodule of M is isomorphic to a direct summand of M and M is called completely virtually semisimple if every submodule of M is virtually semisimple. We show that the left R-module R is completely virtually semisimple if and only if R has a unique decomposition as a finite direct product of matrix rings over principal left ideal domains. This shows that R is completely virtually semisimple on both sides if and only if every finitely generated (left and right) R-module is a direct sum of a singular module and a projective virtually semisimple module. The Wedderburn-Artin theorem follows as a corollary from our result.