Commutative Domain Research Articles

Automorphism groups of geometric lattices

We want to consider automorphism groups of particular lattices. While distributivity of lattices is closely connected with lattices of sets - as noted by G. Birkhoff (see [26]), modular lattices are linked to projective geometry. More precisely, projective geometries are characterized by algebraic, modular lattices that are atomistic and irreducible, see a result of F. and S. Maeda (1970) in [26, p. 539]. We are interested in symmetry groups of generalized projective geometries, hence from a lattice theory point of view our results are about modular lattices. The main feature of our approach is an attempt to apply strong realization theorems for groups and rings as automorphism groups and endomorphism rings respectively, see [7, 8, 9, 10, 11, 32, 33]. These results will be used for an interplay of linear and semi-linear transformations in a setting of projective geometry. One of our main results is an application of a version of the fundamental theorem of projective geometry and a new result on endomorphism rings tailored for lattices. They provide lattices of finitely generated submodules of suitable modules over commutative domains with prescribed automorphism group.

MODULES WHOSE MAXIMAL SUBMODULES HAVE SUPPLEMENTS

A ring R is semiperfect if and only if, for every (cyclic)R-module M, every maximal submodule has (ample) supplements in M. A non-local commutative domain R is h-local if and only if, for every (cyclic) torsion R-module M, every maximal submodule of M has (ample) supplements in M.

Simple holonomic modules over rings of differential operators with regular coefficients of Krull dimension 2

Let K K be an algebraically closed field of characteristic zero. Let Λ \Lambda be the ring of ( K K -linear) differential operators with coefficients from a regular commutative affine domain of Krull dimension 2 2 which is the tensor product of two regular commutative affine domains of Krull dimension 1 1 . Simple holonomic Λ \Lambda -modules are described. Let a K K -algebra D D be a regular affine commutative domain of Krull dimension 1 1 and D ( D ) \mathcal {D} (D) be the ring of differential operators with coefficients from D D . We classify (up to irreducible elements of a certain Euclidean domain) simple D ( D ) \mathcal {D}(D) -modules (the field K K is not necessarily algebraically closed).

Primary modules over commutative rings

The radical of a module over a commutative ring is the intersection of all prime submodules. It is proved that if R is a commutative domain which is either Noetherian or a UFD then R is one-dimensional if and only if every (finitely generated) primary R-module has prime radical, and this holds precisely when every (finitely generated) R-module satisfies the radical formula for primary submodules.

The quotient rings of a class of lattice-ordered Ore domains

Let R be a lattice-ordered ring and a commutative domain. It is shown that if R is algebraic over a subring which is totally ordered and whose positive elements are d-elements of R, then the quotient field of R can be made into a lattice-ordered ring extension of R.

A note on skew differential operators on commutative rings

Let A be a commutative integral domain that is a finitely generated algebra over a field k of characteristic 0 and let ø be a k-algebra automorphism of A of finite order m. In this note we study the ring D(A;ø of differential operators introduced by A.D. Bell. We prove that if A is a free module over the fixed sub-ring A ø, with a basis containing 1, then D(A;ø) is isomorphic to the matrix ring Mm(D(A ø). It follows from Grothendieck's Generic Flatness Theorem that for an arbitrary A there is an element cϵAsuch that D(A[c-1];ø)≅M m(D(A[c-1]ø)). As an application, we consider the structure of D(A;ø)when A is a polynomial or Laurent polynomial ring over k and ø is a diagonalizable linear automorphism.

On the atomic property for power series rings

Let A be a commutative domain. We prove that both implications A atomic ⇒ A[[X]] atomic , and A[[X]] atomic ⇒ A atomic fail in general.

Cantor-Bendixson Rank of the Ziegler Spectrum Over a Commutative Valuation Domain

AbstractWe calculate the Cantor-Bendixson rank of the Ziegler spectrum over a commutative valuation domain R proving that it is equal to the double Krull dimension of R.

On a class of divisors of matrices

We propose a structural criterion for divisibility of matrices over a commutative principal ideal domain. Under certain restrictions we exhibit a method of finding all the non-associate divisors that have a prescribed canonical diagonal form.

A Characterization of Prime Submodules

Let R be a commutative domain and let M be an R-module. It is proved that to every prime submodule of M there corresponds a prime ideal of R and a set of linear equations of a certain type, and conversely. In particular, in case M is a finitely generated R-module generated by n elements, for some positive integer n, then the prime submodules of M are given by prime ideals of R and certain finite systems of equations containing at most n equations.

Equivalence elementaire et decidabilite pour des structures du type groupe agissant sur un groupe abelien

AbstractWe prove an Ax-Kochen-Ershov like transfer principle for groups acting on groups. The simplest case is the following: let B be a soluble group acting on an abelian group G so that G is a torsion-free divisible module over the group ring ℤ[B], then the theory of B determines the one of the two-sorted structure 〈G,B,*〉, where * is the action of B on C. More generally, we show a similar principle for structures 〈G,B,*〉, where G is a torsion-free divisible module over the quotient of ℤ[B] by the annulator of G.Two applications come immediately from this result:First, for not necessarily commutative domains, where we consider the action of a subgroup of the invertible elements on the additive group. We obtain then the decidability of a weakened structure of ring, with partial multiplication.The second application is to pure groups. The semi-direct product of G by B is bi-interpretable with our structure 〈G,B,*〉. Thus, we obtain stable decidable groups that are not linear over a field.

Solving quadratic equations over polynomial rings of characteristic two

We are concerned with solving polynomial equations over rings. More precisely, given a commutative domain $A$ with 1 and a polynomial equation $a_n\,t^n+\cdots +a_0=0$ with coefficients $a_i$ in $A$, our problem is to find its roots in $A$. We show that when $A=B[x]$ is a polynomial ring, our problem can be reduced to solving a finite sequence of polynomial equations over $B$. As an application of this reduction, we obtain a finite algorithm for solving a polynomial equation over $A$ when $A$ is $F[x_1,\ldots,x_N]$ or $F(x_1,\ldots,x_N)$ for any finite field $F$ and any number $N$ of variables. The case of quadratic equations in characteristic two is studied in detail.

Reductions and global dimension of quantized algebras over a regular commutative domain

Reductions and global dimension of quantized algebras over a regular commutative domain

Finite polynomial orbits in finitely generated domains

LetR be a commutative domain of zero characteristic and letf(X) be a polynomial with coefficients inR. It is shown that all finite orbits inR. under the mapping induced byR have their cardinalities bounded by a constant depending onR but not onf. In the case whenR is the ring of all integers in an algebraic number field this constant is effectively determined.

A criterion of diagonalizability of a pair of matrices over the ring of principal ideals by common row and separate column transformations

We establish that a pair A, B, of nonsingular matrices over a commutative domain R of principal ideals can be reduced to their canonical diagonal forms D A and D B by the common transformation of rows and separate transformations of columns. This means that there exist invertible matrices U, V A, and V B over R such that UAV a=DA and UAV B=DB if and only if the matrices B *A and D * B DA where B * 0 is the matrix adjoint to B, are equivalent.

The number of generators for the product of an ideal with its inverse

Let (R,m) be a commutative one-dimensional Noetherian local domain. Assume the integral closure satisfying asymptotic condition where b is a strictly positive C∞ function on the unit sphere in R2. is a discrete valvation ring and the residue fields of R and are the same. Let I be a non-principal ideal of R. A question, motivated by the work of Auslander in the early 1960's, is whether the torsion submodule of I ⊗R I−1 is always non-zero. In this paper we investigate the stronger condition, μR(I)μR(I−1) greater than sign μR(II −1)(where μ denotes the number of generators). We will show this inequality holds whenever the multiplicity of R is at most four.

Primary decompositions over domains

Throughout, R denotes a commutative domain with 1, and Q (≠R) its field of quotients, which is viewed here as an R-module. The symbol K will stand for the R-module Q/R, while R denotes the multiplicative monoid R/0.

Central *-Differential Identities in Prime Rings

AbstractLet R be a prime ring with involution and d, δ be derivations on R. Suppose that xd(x)—δ(x)x is central for all symmetric x or for all skew x. Then d = δ = 0 unless R is a commutative integral domain or an order of a 4-dimensional central simple algebra.

Generalized adequate rings

We introduce a new class of rings of elementary divisors which generalize adequate rings. We show that the problem of whether every commutative Bezout domain is a domain of elementary divisors reduces to the case where the domain contains only trivial adequate elements (namely, the identities of the domain).

Skew *-derivations with power-central values

Let R be a prime ring with involution * and d be a nonzero derivation on R such that d(x *) = -d(x)* for all x ∈ R. Suppose that n ≥ 1 is a fixed integer. Then (I) if d(s) n = 0 for all s = s *, then R is either a commutative domain or an order in a 4-dimensional central simple algebra; (II) if d(s) n ∈Z, the center of R for all s = s *, then R is either a commutative domain or an order in a simple algebra of dimension 4 or 16 over its center.