caa a22 4500 606199640 CHVBK 20210128100939.0 cr unu---uuuuu 210128e20150401xx s 000 0 eng 10.1007/s11045-013-0253-4 doi (NATIONALLICENCE)springer-10.1007/s11045-013-0253-4 Oberst Ulrich Institut für Mathematik, Universität Innsbruck, Technikerstrasse 13, 6020, Innsbruck, Austria aut The computation of purity filtrations over commutative noetherian rings of operators and their application to behaviors [Elektronische Daten] [Ulrich Oberst] Due to the theoretical work and computer implementations of, for instance, Barakat, Quadrat and Robertz and their coauthors the theory of finitely generated (f.g.) modules over non-commutative regular noetherian rings of partial differential operators with variable coefficients like the Weyl algebras and over other similar rings has become constructive in recent years. In particular these authors compute the purity or grade filtration of a f.g. module by homological means and discuss its significance for the associated behavior. Pommaret and Quadrat noted this significance already in 1999. In this note it is shown that over an arbitrary commutative noetherian ring of operators the purity filtration of a finitely generated module can be easily computed by means of the the primary decomposition of its zero submodule and indeed, with smaller complexity, by inductively computing equidimensional parts. In most books on Constructive Commutative Algebra the connection between this primary decomposition, the equidimensional parts and the purity filtration is implicitly stated for cyclic modules. For many commutative rings of operators the standard signal modules are injective cogenerators. In this case the purity filtration of the module gives rise to a corresponding filtration of the dual behavior, and the primary decomposition induces additional sum representations of the pure dimensional factors of this filtration. For non-commutative rings of operators the standard signal modules are in general neither injective nor cogenerators, and for such signal modules the usefulness of the purity filtration of the module for the determination of the behavior and its structural properties is not obvious. It is also shown by a counter-example that dimensional purity of the module or behavior does not imply dimensional purity of the initial conditions according to Riquier of the associated homogeneous Cauchy problem. Springer Science+Business Media New York, 2013 Purity filtration nationallicence Primary decomposition nationallicence Multidimensional behavior filtration nationallicence Riquier's initial condition nationallicence Multidimensional Systems and Signal Processing Springer US; http://www.springer-ny.com 26/2(2015-04-01), 389-404 0923-6082 26:2<389 2015 26 11045 https://doi.org/10.1007/s11045-013-0253-4 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s11045-013-0253-4 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Oberst Ulrich Institut für Mathematik, Universität Innsbruck, Technikerstrasse 13, 6020, Innsbruck, Austria aut NATIONALLICENCE 773 0- Multidimensional Systems and Signal Processing Springer US; http://www.springer-ny.com 26/2(2015-04-01), 389-404 0923-6082 26:2<389 2015 26 11045