19 research outputs found
Sets of lengths in maximal orders in central simple algebras
Let be a holomorphy ring in a global field , and a
classical maximal -order in a central simple algebra over . We
study sets of lengths of factorizations of cancellative elements of 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
, 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 . If is the ring of algebraic integers of a
number field , we prove that in the remaining cases no such transfer
homomorphism can exist and that several invariants dealing with sets of lengths
are infinite.Comment: 40 pages; final version, with minor edits over previous on
On the Davenport constant and group algebras
For a finite abelian group and a splitting field of , let denote the largest integer for which there is a sequence over such that for all . If
denotes the Davenport constant of , then there is the straightforward
inequality . Equality holds for a variety of groups, and a
standing conjecture of W. Gao et.al. states that equality holds for all groups.
We offer further groups for which equality holds, but we also give the first
examples of groups for which holds. Thus we disprove
the conjecture.Comment: 12 pages; fixed typos and clearer proof of Lemma 3.
Factorization theory: From commutative to noncommutative settings
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
-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 upper triangular matrices over a
commutative domain using a weak transfer homomorphism to a commutative
semigroup.Comment: 45 page
Cyclically presented modules, projective covers and factorizations
We investigate projective covers of cyclically presented modules,
characterizing the rings over which every cyclically presented module has a
projective cover as the rings that are Von Neumann regular modulo their
Jacobson radical and in which idempotents can be lifted modulo .
Cyclically presented modules naturally appear in the study of factorizations of
elements in non-necessarily commutative integral domains. One of the possible
applications is to the modules whose endomorphism ring is Von
Neumann regular modulo and in which idempotents lift modulo .Comment: 17 page