Radicals Coinciding with the Von Neumann Regular Radical on Artinian Rings.
The Dynkin and the extended Dynkin graphs are characterized by representations over the real numbers.
Let A be a noetherian local commutative ring and let M be a suitable complex of A-modules. It is proved that M is a dualizing complex for A if and only if the trivial extension A ⋉ M is a Gorenstein differential graded algebra. As a corollary, A has a dualizing complex if and only if it is a quotient of a Gorenstein local differential graded algebra.
We generalize the relative (co)tilting theory of Auslander-Solberg in the category mod Λ of finitely generated left modules over an artin algebra Λ to certain subcategories of mod Λ. We then use the theory (relative (co)tilting theory in subcategories) to generalize one of the main result of Marcos et al. [Comm. Algebra 33 (2005)].
Let be a ring, a fixed non-negative integer, the class of all left -modules with weak injective dimension at most , and the class of all right -modules with weak flat dimension at most . Using left (right) -resolutions and the left derived functors of Hom we study the weak injective dimensions of modules and rings. Also we prove that is right balanced on by , and investigate the global right -dimension of by right derived functors of .
We prove the following result: Theorem. Every algebraic distributive lattice D with at most ℵ1 compact elements is isomorphic to the ideal lattice of a von Neumann regular ring R.(By earlier results of the author, the ℵ1 bound is optimal.) Therefore, D is also isomorphic to the congruence lattice of a sectionally complemented modular lattice L, namely, the principal right ideal lattice of R. Furthermore, if the largest element of D is compact, then one can assume that R is unital, respectively,...
We consider a class of two-dimensional non-commutative Cohen-Macaulay rings to which a Brauer graph, that is, a finite graph endowed with a cyclic ordering of edges at any vertex, can be associated in a natural way. Some orders Λ over a two-dimensional regular local ring are of this type. They arise, e.g., as certain blocks of Hecke algebras over the completion of at (p,q-1) for some rational prime . For such orders Λ, a class of indecomposable maximal Cohen-Macaulay modules (see introduction)...