Let F be characteristic zero field, G a residually finite group and W a G-prime and PI F-algebra. By constructing G-graded central polynomials for W , we prove the G-graded version of Posner's theorem. More precisely, if S denotes all non-zero degree e central elements of W , the algebra S −1 W is G-graded simple and finite dimensional over its center.Furthermore, we show how to use this theorem in order to recapture the result of Aljadeff and Haile stating that two G-simple algebras of finite dimension are isomorphix iff their ideals of graded identities coincide.
We count primitive lattices of rank d inside Z n , as their covolume tends to infinity, w.r.t. certain parameters of such lattices. These parameters include, for example, the direction of a lattice, which is the subsapce that it spans; its homothety class; and its equivalence class modulo rescaling and rotation, often referred to as a shape. We add to a prior work of Schmidt by allowing sets that are general enough in the spaces of parameters to conclude joint equidistribution of these parameters.The main novelty is that, in addition to the primitive d-lattices Λ themselves, we also consider their orthogonal lattices Λ ⊥ and the quotients Z n /Λ. We are able to count not only w.r.t. parameters of the primitive lattices Λ, but also w.r.t. the same parameters of Λ ⊥ and Z n /Λ. By doing so, we achieve joint equidistribution of, e.g., the shapes of primitive lattices, and the shapes of their orthogonal lattices. Finally, our asymptotic formulas for the number of primitive lattices includes an explicit error term.
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