The aim of this paper is to investigate strong notion strongly ⨁-supplemented modules in module theory, namely ⨁-locally artinian supplemented modules. We call a M if it locally and its supplement submodules are direct summand. In study, we provide the basic properties particular, show that every summand supplemented. Moreover, prove ring R semiperfect with radical only projective R-module

There has been much work on strong and weak Lefschetz conditions for graded Artinian algebras A, especially those that are Gorenstein. A more general invariant of an algebra or finite A-module M we consider here is the set Jordan types elements maximal ideal

Ladders were introduced by Igusa and Todorov for the investigation of representation-finite artinian algebras and algebras over an algebraically closed field [7]. They prove a radical layers theorem [7] which exhibits the graded structure of Auslander-Reiten sequences. In a second article [8] they obtain a characterization of the Auslander-Reiten quivers of representation-finite artinian algebr...

Let R be a standard graded Noetherian algebra over an Artinian local ring. Motivated by the work of Achilles and Manaresi in intersection theory, we first express the multiplicity of R by means of local j-multiplicities of various hyperplane sections. When applied to a homogeneous inclusion A ⊆ B of standard graded Noetherian algebras over an Artinian local ring, this formula yields the multipl...

In this paper we compute the deformation theory of a special class of algebras, namely of Azumaya algebras on a manifold (C or complex analytic). Deformation theory of associative algebras was initiated by Gerstenhaber in [G]. A deformation of an associative algebra A over an Artinian ring a is an a-linear associative algebra structure on A⊗ a such that, for the maximal ideal m of a, A ⊗ m is a...