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...

1. Ordered Classes, Isotone Maps, and Length Functions 1 1.1. Isotone Maps and Artinian Ordered Classes 1 1.2. Length Functions 2 2. Euclidean Functions 3 2.1. Basic Definitions 3 2.2. Structure Theory of Principal Rings 4 2.3. Generalized Euclidean Functions 4 2.4. Isotone Euclidean Functions 5 2.5. The Bottom Euclidean Function and the Euclidean Order Type 6 2.6. The Localized Euclidean Funct...

In this paper, we study Lefschetz properties of Artinian reductions of Stanley–Reisner rings of balanced simplicial 3-polytopes. A (d − 1)-dimensional simplicial complex is said to be balanced if its graph is d-colorable. If a simplicial complex is balanced, then its Stanley–Reisner ring has a special system of parameters induced by the coloring. We prove that the Artinian reduction of the Stan...

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...

Chow rings of matroids were instrumental in the resolution Heron–Rota–Welsh Conjecture by Adiprasito, Huh, and Katz Top-Heavy Braden, Matherne, Proudfoot, Wang. The ring a matroid is commutative, graded, Artinian, Gorenstein algebra with linear quadratic relations defined matroid. Dotsenko conjectured that any Koszul. purpose this paper to prove Dotsenko’s conjecture. We also show augmented As ...

It is proved that a commutative ring is clean if and only if it is Gelfand with a totally disconnected maximal spectrum. It is shown that each indecomposable module over a commutative ring R satisfies a finite condition if and only if R P is an artinian valuation ring for each maximal prime ideal P. Commutative rings for which each indecomposable module has a local endomorphism ring are studied...