The Local Duality for Homomorphisms and an Application to Pure Semisimple PI-Rings
We give a new proof of the main result of [1] which does not use the classification of the finite simple groups.
Given a module M over a domestic canonical algebra Λ and a classifying set X for the indecomposable Λ-modules, the problem of determining the vector such that is studied. A precise formula for , for any postprojective indecomposable module X, is computed in Theorem 2.3, and interrelations between various structures on the set of all postprojective roots are described in Theorem 2.4. It is proved in Theorem 2.2 that a general method of finding vectors m(M) presented by the authors in Colloq....
Given a module M over an algebra Λ and a complete set of pairwise nonisomorphic indecomposable Λ-modules, the problem of determining the vector such that is studied. A general method of finding the vectors m(M) is presented (Corollary 2.1, Theorem 2.2 and Corollary 2.3). It is discussed and applied in practice for two classes of algebras: string algebras of finite representation type and hereditary algebras of type . In the second case detailed algorithms are given (Algorithms 4.5 and 5.5).
Let be an Archimedean partially ordered ring in which the square of every element is positive, and the set of all nilpotent elements of . It is shown that is the unique nil radical of , and that is locally nilpotent and even nilpotent with exponent at most when is 2-torsion-free. is without non-zero nilpotents if and only if it is 2-torsion-free and has zero annihilator. The results are applied on partially ordered rings in which every element is expressed as with positive ,...
The Dynkin algebras are the hereditary artin algebras of finite representation type. The paper determines the number of complete exceptional sequences for any Dynkin algebra. Since the complete exceptional sequences for a Dynkin algebra of Dynkin type Δ correspond bijectively to the maximal chains in the lattice of non-crossing partitions of type Δ, the calculations presented here may also be considered as a categorification of the corresponding result for non-crossing partitions.