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The main purpose of this paper is to study when a ( T, I, D) construction ring is a C-ring in the sense of Echi (Port Math 50:277–295, 1993) where I is not necessarily a maximal ideal of T. This study allows us to generalize some results proved in Echi (Port Math 50:13–23, 1994) and to give new examples of C-rings. [ABSTRACT FROM AUTHOR]

We present a mathematical method for identifying, separating and quantifying the 3D significant distortions. Measurement of a gas quenched C-ring type sample is performed by a Coordinate Measuring Machine (CMM). Quenching simulation is done with the commercial software Forge 2008 TTT®. After comparisons, we notice that the distortions and their tendencies are the same but not exactly the amplitudes. We only focus on distortions which have a physical origin, like the pincers opening, often observed in the literature. As a first explanation, we underline the role of phase transformations and volume dilatation of the steel during quenching. [ABSTRACT FROM AUTHOR]

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 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 Galois module is introduced and the structure theorem, generalising Schneider’s Theorem II [Schneider, Isr. J. Math., 72:167–195, 1990], 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 infinite matrix ring contexts) is outlined. A Galois connection associated to a matrix C-ring is constructed. [ABSTRACT FROM AUTHOR]