A classical result of Claborn states that every abelian group is the class group of a commutative Dedekind domain. Among noncommutative Dedekind prime rings, apart from PI rings, the simple Dedekind ...domains form a second important class. We show that every abelian group is the class group of a noncommutative simple Dedekind domain. This solves an open problem stated by Levy and Robson in their recent monograph on hereditary Noetherian prime rings.
Definite orders with locally free cancellation Smertnig, Daniel; Voight, John
Transactions of the London Mathematical Society,
December 2019, 2019-12-00, 20191201, 2019-12-01, Letnik:
6, Številka:
1
Journal Article
Recenzirano
Odprti dostop
We enumerate all orders in definite quaternion algebras over number fields with the Hermite property; this includes all orders with the cancellation property for locally free modules.
Let O be a holomorphy ring in a global field K, and R a classical maximal O-order in a central simple algebra over K. We study sets of lengths of factorizations of cancellative elements of R into ...atoms (irreducibles). In a large majority of cases there exists a transfer homomorphism to a monoid of zero-sum sequences over a ray class group of O, which implies that all the structural finiteness results for sets of lengths—valid for commutative Krull monoids with finite class group—hold also true for R. If O is the ring of algebraic integers of a number field K, we prove that in the remaining cases no such transfer homomorphism can exist and that several invariants dealing with sets of lengths are infinite.
If H is a monoid and a = u1 ··· uk ∈ H with atoms (irreducible elements) u1, … , uk, then k is a length of a, the set of lengths of a is denoted by Ⅼ(a), and ℒ(H) = {Ⅼ(a) | a ∈ H} is the system of ...sets of lengths of H. Let R be a hereditary Noetherian prime (HNP) ring. Then every element of the monoid of non-zero-divisors R• can be written as a product of atoms. We show that if R is bounded and every stably free right R-ideal is free, then there exists a transfer homomorphism from R• to the monoid B of zero-sum sequences over a subset Gmax(R) of the ideal class group G(R). This implies that the systems of sets of lengths, together with further arithmetical invariants, of the monoids R• and B coincide. It is well known that commutative Dedekind domains allow transfer homomorphisms to monoids of zero-sum sequences, and the arithmetic of the latter has been the object of much research. Our approach is based on the structure theory of finitely generated projective modules over HNP rings, as established in the recent monograph by Levy and Robson. We complement our results by giving an example of a non-bounded HNP ring in which every stably free right R-ideal is free but which does not allow a transfer homomorphism to a monoid of zero-sum sequences over any subset of its ideal class group.
We study the non-uniqueness of factorizations of non zero-divisors into atoms (irreducibles) in noncommutative rings. To do so, we extend concepts from the commutative theory of non-unique ...factorizations to a noncommutative setting. Several notions of factorizations as well as distances between them are introduced. In addition, arithmetical invariants characterizing the non-uniqueness of factorizations such as the catenary degree, the ω-invariant, and the tame degree, are extended from commutative to noncommutative settings. We introduce the concept of a cancellative semigroup being permutably factorial, and characterize this property by means of corresponding catenary and tame degrees. Also, we give necessary and sufficient conditions for there to be a weak transfer homomorphism from a cancellative semigroup to its reduced abelianization. Applying the abstract machinery we develop, we determine various catenary degrees for classical maximal orders in central simple algebras over global fields by using a natural transfer homomorphism to a monoid of zero-sum sequences over a ray class group. We also determine catenary degrees and the permutable tame degree for the semigroup of non zero-divisors of the ring of n×n upper triangular matrices over a commutative domain using a weak transfer homomorphism to a commutative semigroup.
A multivariate, formal power series over a field K is a Bézivin series if all of its coefficients can be expressed as a sum of at most r elements from a finitely generated subgroup
$G \le K^*$
; it ...is a Pólya series if one can take
$r=1$
. We give explicit structural descriptions of D-finite Bézivin series and D-finite Pólya series over fields of characteristic
$0$
, thus extending classical results of Pólya and Bézivin to the multivariate setting.
Noncommutative rational Pólya series Bell, Jason; Smertnig, Daniel
Selecta mathematica (Basel, Switzerland),
07/2021, Letnik:
27, Številka:
3
Journal Article
Recenzirano
Odprti dostop
A (noncommutative) Pólya series over a field
K
is a formal power series whose nonzero coefficients are contained in a finitely generated subgroup of
K
×
. We show that rational Pólya series are ...unambiguous rational series, proving a 40 year old conjecture of Reutenauer. The proof combines methods from noncommutative algebra, automata theory, and number theory (specifically, unit equations). As a corollary, a rational series is a Pólya series if and only if it is Hadamard sub-invertible. Phrased differently, we show that every weighted finite automaton taking values in a finitely generated subgroup of a field (and zero) is equivalent to an unambiguous weighted finite automaton.
In J. reine angew. Math. 286/287 (1976), 257–277; J. reine angew. Math. 595 (2006), 189–213 it is shown that there exist only finitely many isomorphism classes of Eichler orders of square-free level ...in totally definite quaternion algebras over number fields having locally free cancellation, and they are all classified. There is an error in this classification which is corrected in the present note.
The (left) linear hull of a weighted automaton over a field is a topological invariant. If the automaton is minimal, the linear hull can be used to determine whether or not the automaton is ...equivalent to a deterministic one. Furthermore, the linear hull can also be used to determine whether the minimal automaton is equivalent to an unambiguous one. We show how to compute the linear hull, and thus prove that it is decidable whether or not a given automaton over a number field is equivalent to a deterministic one. In this case we are also able to compute an equivalent deterministic automaton. We also show the analogous decidability and computability result for the unambiguous case. Our results resolve a problem posed in a 2006 survey by Lombardy and Sakarovitch.
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