Let X be a codimension 1 subvariety of dimension > 1 of a variety of minimal degree Y . If X is subcanonical with Gorenstein canonical singularities admitting a crepant resolution, then X is Arithmetically Gorenstein and we characterise such subvarieties X of Y via apolarity as those whose apolar hypersurfaces are Fermat.

we define and studyco-noetherian dimension of rings for which the injective envelopeof simple modules have finite krull-dimension. this  is a moritainvariant dimension that measures how far the ring is from beingco-noetherian. the co-noetherian dimension of certain rings,including commutative rings, are determined. it is shown that the class ${mathcal w}_n$ of rings with co-noetherian dimension...

Let (R, m, K) be a Gorenstein local ring of prime characteristic p > 0, and let x1, . . . , xn be a system of parameters. Let It = (x1, . . . , x t n)R for all t ≥ 1, and let u ∈ R represent a socle generator in R/I, where I = I1 = (x1, . . . , xn)R. Let y = x1 · · · xn. We have seen that E = lim −→ t R/It is an injective hull of K = R/m over R, where the map R/It → R/It+1 is induced by multipl...