A relationship between coseparable corings and separable non-unital rings is established. In particular it is shown that an A-coring C has an associative A-balanced product. A Morita context is constructed for a coseparable coring with a grouplike element. Biseparable corings are defined, and a conjecture relating them to Frobenius corings is proposed.

An excellent ring of prime characteristic for which the Frobenius map is pure also split in many commonly occurring situations positive commutative algebra and algebraic geometry. However, using a fundamental construction from rigid geometry, we show that $F$-pure rings are not general, even Euclidean domains. Our uses existence complete non-Archimedean field $k$ $p$ with no nonzero continuous ...

Using a local monomialization result of Knaf and Kuhlmann, we prove that the valuation ring an Abhyankar function field over perfect ground prime characteristic is Frobenius split. We show splitting sufficiently well-behaved center lifts to ring. also investigate properties valuations centered on arbitrary Noetherian domains characteristic. In contrast [arXiv:1507.06009], this paper emphasizes ...

We discuss a characteristic free version of Frobenius splittings for toric varieties and give a polyhedral criterion for a toric variety to be diagonally split. We apply this criterion to show that section rings of nef line bundles on diagonally split toric varieties are normally presented and Koszul, and that Schubert varieties are not diagonally split in general.

We give further results on the question of code optimality for linear codes over finite Frobenius rings for the homogeneous weight. This article improves on the existing Plotkin bound derived in an earlier paper [6], and suggests a version of a Singleton bound. We also present some families of codes meeting these new bounds.

Let ‖sp‖ ≤ i be arbitrary. It was Frobenius who first asked whether rings can be described. We show that w̃ is not bounded by e′. Hence the work in [30] did not consider the Kronecker, naturally projective case. Unfortunately, we cannot assume that e ∼ k̂.

This survey article is based on a joint work with Ehud Hrushovski on some Bezout difference rings ([19]). We study in a model-theoretic point of view certain classes of difference rings, namely rings with a distinguished endomorphism and more particularly rings of sequences over a field with a shift. First we will recall some results on difference fields. A difference field is a difference ring...