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 $Q_{\mathrm{tot}}^r(R),$ 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 $R$. In [K. Morita, Flat modules, injective modules and quotient rings, Math. Z. 120 (1971) 25--40], $Q_{\mathrm{tot}}^r(R)$ is constructed in a different way, by transfinite induction on ordinals. Starting with the maximal right ring of quotients $Q_{\mathrm{max}}^r(R)$, its subrings are constructed until $Q_{\mathrm{tot}}^r(R)$ is obtained. Here, we prove that Morita's construction of $Q_{\mathrm{tot}}^r(R)$ can be simplified for rings satisfying condition (C) that every subring of the maximal right ring of quotients $Q^r_{\mathrm{max}}(R)$ containing $R$ is flat as a left $R$-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 $Q^r_{\mathrm{tot}}(R)$ 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 ${\mathcal C}$ 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 $R$ satisfies the first seven axioms, is the matrix ring $M_n(R)$ a Baer *-ring

Semisimplicity and global dimension of a finite von Neumann algebra

We prove that a finite von Neumann algebra ${\mathcal A}$ is semisimple if the algebra of affiliated operators ${\mathcal U}$ of ${\mathcal A}$ is semisimple. When ${\mathcal A}$ is not semisimple, we give the upper and lower bounds for the global dimensions of ${\mathcal A}$ and ${\mathcal U}.$ This last result requires the use of the Continuum Hypothesis