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We introduce a new intrinsic notion of spread for toposes and geometric morphisms, and use it to give a “topological” characterization of Lawvere distributions on a topos. In the process, we relate spreads to zero-dimensional locales, and establish two new pure/spread factorizations for...
We show that over any left distributive or left duo ring R there exists a 1-1 correspondence between ∑-injective indecomposable left modules and completely prime ideals P such that the left classical localization R ( P ) exists and is a left noetherian ring. Notice that the well-known theorems...
It is well-known that weak approximation holds for a large class of semisimple groups over global fields, including those which are simply connected or adjoint. Earlier Kneser suggested the investigation of weak approximation in algebraic groups over any field of definition and Platonov gave...
Given an integral vector u / gE Z n , one may associate with it the binomial / tf u = X u + − X u − in Z ( X ) = Z ( X 1 , …, X n ) where u + and u − are the positive and negative supports of u , respectively. We say that u is mixed if u + , u − ≠ 0 and a matrix M is mixed if all its...
Let R = k ( x 1 , …, x n ) and R ( x ) be a polynomial ring over a field k and let I be a normal ideal of R generated by square free monomials of the same degree. We prove that I + x ( x 1 , …, x n ) and I + ( xx 1 ) are both normal ideals of R ( x ). The ideals I t , and I t + xI t − 2 are...
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