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Register a new userA Note on a Standard Model for Galois Rings
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In this work, we present a standard model for Galois rings based on the standard model of their residual fields, that is, a sequence of Galois rings starting with ${mathbb Z}_{p^r} that coves all the Galois rings with that characteristic ring and such that there is an algorithm producing each member of the sequence whose input is the size of the required ring.
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A theory of monoids in the category of bicomodules of a coalgebra $C$ or $C$-rings is developed. This can be viewed as a dual version of the coring theory. The notion of a matrix ring context consisting of two bicomodules and two maps is introduced and the corresponding example of a $C$-ring (termed a {em matrix $C$-ring}) is constructed. It is shown that a matrix ring context can be associated to any bicomodule which is a one-sided quasi-finite injector. Based on this, the notion of a {em Galois module} is introduced and the structure theorem, generalising Schneiders Theorem II [H.-J. Schneider, Israel J. Math., 72 (1990), 167--195], is proven. This is then applied to the $C$-ring associated to a weak entwining structure and a structure theorem for a weak $A$-Galois coextension is derived. The theory of matrix ring contexts for a firm coalgebra (or {em infinite matrix ring contexts}) is outlined. A Galois connection associated to a matrix $C$-ring is constructed.
The recently developed theory of Schur rings over a finite cyclic group is generalized to Schur rings over a ring R being a product of Galois rings of coprime characteristics. It is proved that if the characteristic of R is odd, then as in the cyclic group case any pure Schur ring over R is the tensor product of a pure cyclotomic ring and Schur rings of rank 2 over non-fields. Moreover, it is shown that in contrast to the cyclic group case there are non-pure Schur rings over R that are not generalized wreath products.
The results of [I. Ojeda, Amer. Math. Monthly, 122, pp 60--64] provides a characterization of Kronecker square roots of matrices in terms of the symmetry and rank of the block vec matrix (rearrangement matrix). In this short note we reformulate the characterization in terms of rank only by considering an alternative to the block vec matrix, provided that the characteristic of the underlying field is not equal to 2.
In this note, we show that a strongly $phi$-ring $R$ is a $phi$-PvMR if and only if any $phi$-torsion free $R$-module is $phi$-$w$-flat, if and only if any divisible module is nonnil-absolutely $w$-pure module, if and only if any $h$-divisible module is nonnil-absolutely $w$-pure module, if and only if any finitely generated nonnil ideal of $R$ is $w$-projective.
We provide a formula for commputing the discriminant of skew Calabi-Yau algebra over a central Calabi-Yau algebra. This method is applied to study the Jacobian and discriminant for reflection Hopf algebras.
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