The Jacobson-Bourbaki Theorem for division rings was formulated in terms of corings by Sweedler in [14]. Finiteness conditions hypotheses are not required in this new approach. In this paper we extend Sweedler’s result to simple artinian rings using a particular class of corings, comatrix corings. A Jacobson-Bourbaki like correspondence for simple artinian rings is then obtained by duality.

We give a characterization of the Lefschetz elements in Artinian Gorenstein rings over a field of characteristic zero in terms of the higher Hessians. As an application, we give new examples of Artinian Gorenstein rings which do not have the strong Lefschetz property.

We investigate two invariants of Noetherian semiperfect rings, namely the depth and a new invariant we call “delooping level”. These give lower upper bounds for finitistic dimension, respectively. As first theorems, necessary sufficient criterion delooping level to be finite in terms splitting unit map related adjunction, use this torsionfreeness finiteness. further relate these Auslander–Bridg...

We'll review the study of Galois deformations. Here's the setup. Let G be a pro nite group, and let ρ : G→ GLn(k) be a representation of G over a nite eld k of characteristic p. Let Λ be a complete DVR with residue eld k, and let CΛ denote the category whose objects are artinian local Λ-algebras with residue eld k, and whose morphisms are local homomorphisms. Let ĈΛ denote the category of compl...

Inspired by a recent paper due to José Luis García, we revisit the attempt of Daniel Simson construct counterexample pure semisimplicity conjecture. Using compactness, show that existence such would readily follow from very certain (countable set of) hereditary artinian rings finite representation type. The is then proved be equivalent special types embeddings, which call tight, division into s...

‎Let $I$ be an ideal in a regular local ring $(R,n)$‎, ‎we will find‎ ‎bounds on the first and the last Betti numbers of‎ ‎$(A,m)=(R/I,n/I)$‎. ‎if $A$ is an Artinian ring of the embedding‎ ‎codimension $h$‎, ‎$I$ has the initial degree $t$ and $mu(m^t)=1$‎, ‎we call $A$ a {it $t-$extended stretched local ring}‎. ‎This class of‎ ‎local rings is a natural generalization of the class of stretched ...

If R is a commutative ring, then we prove that every finitely generated R-module has a pure-composition series with indecomposable cyclic factors and any two such series are isomorphic if and only if R is a Bézout ring and a CF-ring. When R is a such ring, the length of a pure-composition series of a finitely generated R-module M is compared with its Goldie dimension and we prove that these num...

A right Johns ring is a Noetherian in which every ideal annihilator. It known that RR the Jacobson radical J(R)J(R) of nilpotent and Soc(R)(R) an essential RR. Moreover, Kasch, is, simple RR-module can be embedded For M∈RM∈R-Mod we use concept MM-annihilator define module (resp. quasi-Johns) as MM such submodule MM-annihilator. called quasi-Johns if any set submodules satisfies ascending chain ...