33,017 research outputs found
Formalized linear algebra over Elementary Divisor Rings in Coq
This paper presents a Coq formalization of linear algebra over elementary
divisor rings, that is, rings where every matrix is equivalent to a matrix in
Smith normal form. The main results are the formalization that these rings
support essential operations of linear algebra, the classification theorem of
finitely presented modules over such rings and the uniqueness of the Smith
normal form up to multiplication by units. We present formally verified
algorithms computing this normal form on a variety of coefficient structures
including Euclidean domains and constructive principal ideal domains. We also
study different ways to extend B\'ezout domains in order to be able to compute
the Smith normal form of matrices. The extensions we consider are: adequacy
(i.e. the existence of a gdco operation), Krull dimension and
well-founded strict divisibility
Star Stable Domains
We introduce and study the notion of -stability with respect to a
semistar operation defined on a domain ; in particular we consider
the case where is the -operation. This notion allows us to
generalize and improve several properties of stable domains and totally
divisorial domains.Comment: 21 pages. J. Pure Appl. Algebra, to appea
An alternative perspective on projectivity of modules
Similar to the idea of relative projectivity, we introduce the notion of
relative subprojectivity, which is an alternative way to measure the
projectivity of a module. Given modules and , is said to be {\em
-subprojective} if for every epimorphism and
homomorphism , then there exists a homomorphism such that . For a module , the {\em subprojectivity
domain of } is defined to be the collection of all modules such that
is -subprojective. A module is projective if and only if its subprojectivity
domain consists of all modules. Opposite to this idea, a module is said to
be {\em subprojectively poor}, or {\em -poor} if its subprojectivity domain
is as small as conceivably possible, that is, consisting of exactly the
projective modules. Properties of subprojectivity domains and -poor modules
are studied. In particular, the existence of an -poor module is attained
for artinian serial rings.Comment: Dedicated to the memory of Francisco Raggi; v2 some editorial
changes. 'Right hereditary right perfect' replaced by the (equivalent)
condition 'right hereditary semiprimary'; v3 a mistake corrected in the
statements of Propositions 3.8 and 3.
w-Divisorial Domains
We study the class of domains in which each w-ideal is divisorial, extending
several properties of divisorial and totally divisorial domains to a much wider
class of domains. In particular we consider PvMDs and Mori domains
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