73 citations · 300 across the 27 of their papers we have counts for
11 papers · 1 filter
Divide and Transfer: Non-Unique Factorizations Beyond Commutativity
Daniel Smertnig
Unique factorization fails in many rings and monoids, but divisor and transfer homomorphisms provide tools to understand non-unique factorizations. In this expository article, we f…
A monoid-theoretical approach to infinite direct-sum decompositions of modules
Zahra Nazemian, Daniel Smertnig
Let be a class of modules over a ring , closed under direct sums over index sets of cardinality and isomorphisms, and such that the isomorphism classes form a s…
On noncommutative bounded factorization domains and prime rings
Jason P. Bell, Ken Brown, Zahra Nazemian +1
A ring has bounded factorizations if every cancellative nonunit can be written as a product of atoms and there is a bound on the lengths of such factorizations. Th…
On basic and Bass quaternion orders
Sara Chari, Daniel Smertnig, John Voight
A quaternion order O over a Dedekind domain R is Bass if every R-superorder is Gorenstein, and O is basic if it contains an integrally closed quadratic R-order. In this article, we…
Arithmetical invariants of local quaternion orders
Nicholas R. Baeth, Daniel Smertnig
Let be a DVR, let be its quotient field, and let be a -order in a quaternion algebra over . The elasticity of is $ρ(R^\bullet) = \sup\{\, k/l : u_…
Factorizations in bounded hereditary Noetherian prime rings
Daniel Smertnig
If is a monoid and with atoms (irreducible elements) , then is a length of , the set of lengths of is denoted by $\mathsf…
Factorizations of Elements in Noncommutative Rings: A Survey
Daniel Smertnig
We survey results on factorizations of non zero-divisors into atoms (irreducible elements) in noncommutative rings. The point of view in this survey is motivated by the commutative…
Every abelian group is the class group of a simple Dedekind domain
Daniel Smertnig
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, t…
Factorization theory: From commutative to noncommutative settings
Nicholas R. Baeth, Daniel Smertnig
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 no…
Cyclically presented modules, projective covers and factorizations
Alberto Facchini, Daniel Smertnig, Nguyen Khanh Tung
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…
Sets of lengths in maximal orders in central simple algebras
Daniel Smertnig
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 factor…