Derived decompositions of abelian categories are introduced in internal terms subcategories to construct semi-orthogonal (or Bousfield localizations, or hereditary torsion pairs) various derived categories. We give a sufficient condition for arbitrary have such and show that it is also necessary with enough projectives injectives. For bounded categories, we describe which determined by decompos...

we show  that a projective maximal submodule of afinitely generated, regular, extending module is a directsummand. hence, every finitely generated, regular, extendingmodule with projective maximal submodules is semisimple. as aconsequence, we observe that every regular, hereditary, extendingmodule is semisimple. this generalizes and simplifies a result of  dung and   smith. as another consequen...

We prove some general estimates for exponential sums over subsets of finite fields which are definable in the language of rings. This generalizes both the classical exponential sum estimates over varieties over finite fields due to Weil, Deligne and others, and the result of Chatzidakis, van den Dries and Macintyre concerning the number of points of those definable sets. As a first application,...

We study leader election in unidirectional rings of homonym processes that have no a priori knowledge on the number of processes. We show that message-terminating leader election is impossible for any class of rings Kk with bounded multiplicity k ≥ 2. However, we show that process-terminating leader election is possible in the subclass U∗ ∩ Kk, where U ∗ is the class of rings which contain a pr...