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.

By means of a kind of new idea, we redefine some kinds of fuzzy ideals in a ring and investigate some of their related properties. The concepts of strong prime (semiprime) generalized fuzzy (bi-, quasi-) ideals in rings are introduced. In particular, we discuss the relationships between strong prime (resp., semiprime) generalized fuzzy (bi-, quasi-) ideals and prime (resp. semiprime) generalize...

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 ...

Nakayama (Ann. of Math. 42, 1941) showed that over an artinian serial ring every module is a direct sum of uniserial modules. Hence artinian serial rings have the property that each right (left) ideal is a finite direct sum of quasi-injective right (left) ideals. A ring with the property that each right (left) ideal is a finite direct sum of quasi-injective right (left) ideals will be called a ...

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.