Extending Ring Derivations to Right and Symmetric Rings and Modules of Quotients

Abstract

We define and study the symmetric version of differential torsion theories. We prove that the symmetric versions of some of the existing results on derivations on right modules of quotients hold for derivations on symmetric modules of quotients. In particular, we prove that the symmetric Lambek, Goldie, and perfect torsion theories are differential.We also study conditions under which a derivation on a right or symmetric module of quotients extends to a right or symmetric 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.

A Note on (α, β)-higher Derivations and their Extensions to Modules of Quotients

“…In Section 5, we show that the results from previous sections hold for symmetric filters as well (Corollary 5.1). Lastly, in Section 6, we present an example of a torsion theory that is not differential (Example 6) thus answering a question from [8]. Using result from Section 3, we also show that there cannot exist a hereditary torsion theory that is differential but not higher differential.…”

“…As a consequence, we obtain that every Gabriel filter is higher differential (Corollary 3.2) and that every higher derivation on a module extends to its module of quotients (Corollary 3.3). In Section 4, we show that the assumptions for some results from [8] and [9] can be relaxed and that these results hold for every two filters F 1 and F 2 such that F 1 ⊆ F 2 (Corollary 4.1). In Section 5, we show that the results from previous sections hold for symmetric filters as well (Corollary 5.1).…”

“…In particular, the extension of D on module of quotients with respect Lambek or Goldie torsion theory agree with the extension of D with respect to any other hereditary and faithful torsion theory.Proof. By Proposition 3 from[8], if a ∆-HD D extends to M F1 and M F2 , then the extensions agree. By Corollary 3.3, D always extends to both M F1 and M F2 and so the result follows.…”

“…Let d be a derivation on M that extends to M F1 . Conditions under which d can be extended to M F2 so that the extension agree were studied in [9].…”

“…In [9], the concept of invariant filters is extended to symmetric filters as well. A symmetric filter l F r induced by a left filter F l and a right filter F r can be defined so that the hereditary torsion theory l τ r on R-bimodules that correspond to l F r has the torsion class equal to the intersection of torsion classes of τ l and τ r :…”