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

Developing Mathematical Reflektive Thinki NG Skills Through Problem Based Learning

Reflective thinking gives the opportunity to students the chance to assess believe. That means providing relevant information on the student’s belief and reflecting students' understanding of a given topic. In other words, reflective thinking provides the opportunity for student to solve a problem together with the reasons that logically, defend their opinions, analyze and reflect them. A process that facilitates student to re-think / think back when responding or choosing solutions that are useful in developing reflective thinking skills, that is a learning process which can encourage reflective thinking. One of which is a model of learning that can minimize these problems is by using problem based learning. Key Word: Reflektive thinking skills, Problem based learnin

Solving the Mystery of Intrusive Flashbacks in Posttraumatic Stress Disorder (Comment on Brewin 2014)

"This article may not exactly replicate the final version published in the APA journal. It is not the copy of record."In the light of current controversy about the nature of intrusions in posttraumatic stress disorder (PTSD), the review by Brewin (2014) is timely and important. It will undoubtedly stimulate further research and guide researchers' quests for understanding the nature of flashbacks in PTSD. In this commentary, I briefly summarize and discuss key points made by Brewin and elaborate on some of the reasons behind the controversy. For example, the terms involuntary autobiographical memories, intrusive memories, and flashbacks are often used interchangeably. I propose a taxonomy revealing the key differences across these forms of memory. If flashbacks are characteristic of patients with PTSD only, it is essential that more research targeting this population is conducted with a variety of methods. Finally, some new avenues for research to study intrusive memories and flashbacks in PTSD, using a diary method and modified trauma film paradigm, are described.Peer reviewedSubmitted Versio

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