Lia Vas

Lia Vas

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Perfect torsion theories have the desirable feature that every module of quotients is determined solely by the right ring of quotients. On the other hand, symmetric rings of quotients have a symmetry that mimics the commutative case. In this talk, we study rings of quotients that combine these two desirable properties.

Abstract: We consider a class of Baer *-rings 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 tothe 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.

The total right ring of quotients $Q^r_{tot}(R)$ is usually obtained as a directed union of a certain family of extensions of the base ring $R.$ Morita constructs $Q^r_{tot}(R)$ in a different way, starting from the maximal right ring of quotients $Q^r_{max}(R)$ and shrinking it using the transfinite induction on ordinals. We consider some classes of rings for which Morita's construction can be simplified. Using this simplification, we prove that Morita's construction of $Q^r_{tot}(R)$ ends after just one step if $R$ is a right semihereditary ring, producing a hands-on description of $Q^r_{tot}(R)$ in this case.

A torsion theory is called differential if a derivation can be extended from any module to the module of quotients corresponding to the torsion theory. We prove that some widely used torsion theories (the Lambek, Goldie and any perfect torsion theories) are differential. We also study conditions under which a derivation on a module of quotients extends to a module of quotients with respect to a larger torsion theory. Using these results, we study extensions of ring derivations to maximal, total and perfect right and symmetric rings of quotients.

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