Separable rings

Separable Algebras over Commutative Rings

Introduction. The main objects of study in this paper are the commutative separable algebras over a commutative ring. Noncommutative separable algebras have been studied in [2]. Commutative separable algebras have been studied in [1] and in [2], [6] where the main ideas are based on the classical Galois theory of fields. This paper depends heavily on these three papers and the reader should con...

On Separable Abelian Extensions of Rings

Let R be a ring with I, G (= (1>x...Xm) a finite abelian automorphism group of R of order n where (i2 is cyclic of order n. for some integers n, hi, and m, and C the center of R whose automorphism group induced by G is isomorphic with G. Then an abelian extension Rtx 1,...,xm is defined as a generalization of cyclic extensions of rings, and RXl,...,Xm is an Azumaya algebra over K (= CG {c in C ...

Separable Partitions

An ordered partition of a set of n points in the d dimensional Euclidean space is called a separable partition if the convex hulls of the parts are pairwise disjoint. For each fixed p and d we determine the maximum possible number rp,d(n) of separable partitions into p parts of n points in real d-space up to a constant factor. Of particular interest are the values rp,d(n) = Θ(nd( p 2)) for ever...

Differential Polynomial Rings of Triangular Matrix Rings

Separable and non-separable data representation for pattern discrimination

We provide a complete work-flow, based on the language of quantum information theory, suitable for processing data for the purpose of pattern recognition. The main advantage of the introduced scheme is that it can be easily implemented and applied to process real-world data using modest computation resources. At the same time it can be used to investigate the difference in the pattern recogniti...