3,837 research outputs found
Extending higher derivations to rings and modules of quotients
A torsion theory is called differential (higher differential) if a derivation
(higher derivation) can be extended from any module to the module of quotients
corresponding to the torsion theory. We study conditions equivalent to higher
differentiability of a torsion theory. It is known that the Lambek, Goldie and
any perfect torsion theories are differential. We show that these classes of
torsion theories are higher differential as well. Then, we study conditions
under which a higher derivation extended to a right module of quotients extends
also to a right module of quotients with respect to a larger torsion theory.
Lastly, we define and study the symmetric version of higher differential
torsion theories. We prove that the symmetric versions of the results on higher
differential (one-sided) torsion theories hold for higher derivations on
symmetric modules of quotients. In particular, we prove that the symmetric
Lambek, Goldie and any perfect torsion theories are higher differential
A simplification of Morita's construction of total right rings of quotients for a class of rings
The total right ring of quotients sometimes also
called the maximal flat epimorphic right ring of quotients or right flat
epimorphic hull, is usually obtained as a directed union of a certain family of
extension of the base ring . In [K. Morita, Flat modules, injective modules
and quotient rings, Math. Z. 120 (1971) 25--40], is
constructed in a different way, by transfinite induction on ordinals. Starting
with the maximal right ring of quotients , its subrings
are constructed until is obtained.
Here, we prove that Morita's construction of can be
simplified for rings satisfying condition (C) that every subring of the maximal
right ring of quotients containing is flat as a
left -module. We illustrate the usefulness of this simplification by
considering the class of right semihereditary rings all of which satisfy
condition (C). We prove that the construction stops after just one step and we
obtain a simple description of in this case. Lastly, we
study conditions that imply that Morita's construction ends in countably many
steps
Knowledge representation and evaluation an ontology-based knowledge management approach
Competition between Higher Education Institutions is increasing at an alarming rate, while changes of
the surrounding environment and demands of labour market are frequent and substantial. Universities
must meet the requirements of both the national and European legislation environment. The Bologna
Declaration aims at providing guidelines and solutions for these problems and challenges of European
Higher Education. One of its main goals is the introduction of a common framework of transparent and
comparable degrees that ensures the recognition of knowledge and qualifications of citizens all across the
European Union. This paper will discuss a knowledge management approach that highlights the importance
of such knowledge representation tools as ontologies. The discussed ontology-based model supports the
creation of transparent curricula content (Educational Ontology) and the promotion of reliable knowledge
testing (Adaptive Knowledge Testing System)
Class of Baer *-rings Defined by a Relaxed Set of Axioms
We consider a class of Baer *-rings (also treated in [S. K.
Berberian, Baer *-rings, Die Grundlehren der mathematischen Wissenschaften 195,
Springer-Verlag, Berlin-Heidelberg-New York, 1972.] and [L. Va\v{s}, Dimension
and Torsion Theories for a Class of Baer *-Rings, Journal of Algebra 289 (2005)
no. 2, 614--639]) defined by nine axioms, the last two of which are
particularly strong. We prove that the ninth axiom follows from the first
seven. This gives an affirmative answer to the question of S. K. Berberian if a
Baer *-ring satisfies the first seven axioms, is the matrix ring a
Baer *-ring
Semisimplicity and global dimension of a finite von Neumann algebra
We prove that a finite von Neumann algebra is semisimple if
the algebra of affiliated operators of is
semisimple. When is not semisimple, we give the upper and lower
bounds for the global dimensions of and This
last result requires the use of the Continuum Hypothesis
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