In a recent paper [17] Miro-Roig, Mezzetti and Ottaviani highlight the link between rational varieties satisfying a Laplace equation and artinian ideals failing the Weak Lefschetz Property. Continuing their work we extend this link to the more general situation of artinian ideals failing the Strong Lefschetz Property. We characterize the failure of the SLP (which includes WLP) by the existence ...

We obtain a complete structural characterization of Cohn-Leavitt algebras over no-exit objects as graded involutive algebras. Corollaries of this result include graph-theoretic conditions characterizing when a Leavitt path algebra is a directed union of (graded) matricial algebras over the underlying field and over the algebra of Laurent polynomials and when the monoid of isomorphism classes of...

In [17] the authors highlight the link between rational varieties satisfying a Laplace equation and artinian ideals that fail the Weak Lefschetz property. Continuing their work we extend this link to the more general situation of artinian ideals failing the Strong Lefschetz Property. We characterize the failure of SLP (that includes WLP) by the existence of special singular hypersurfaces (cones...

Since the 1970’s, great interest has been taken in the study of pure O-sequences, which, due to Macaulay’s theory of inverse systems, have a bijective correspondence to the Hilbert functions of Artinian level monomial algebras. Much progress has been made in classifying these according to their shape. Macaulay’s theorem immediately gives us that all Artinian algebras in two variables have unimo...

An SI-sequence is a finite sequence of positive integers which is symmetric, unimodal and satisfies a certain growth condition. These are known to correspond precisely to the possible Hilbert functions of Artinian Gorenstein algebras with the Weak Lefschetz Property, a property shared by most Artinian Gorenstein algebras. Starting with an arbitrary SI-sequence, we construct a reduced, arithmeti...

Finite Frobenius rings have been characterized as precisely those finite rings satisfying the MacWilliams extension property, by work of Wood. In the present note we offer a generalization of this remarkable result to the realm of Artinian rings. Namely, we prove that a left Artinian ring has the left MacWilliams property if and only if it is left pseudo-injective and its finitary left socle em...

Let R = ⊕ n∈N0 Rn be a Noetherian homogeneous ring with local base ring (R0,m0) and irrelevant ideal R+, let M be a finitely generated graded R− module. In this paper we show that if R0 is a local ring of dimension one, then H i R+(H 1 m0R (M)) is Artinian for each i ∈ N0. Let f be the least integer such that H i m0R(M) is not finitely generated graded R−module. In this case, we prove that ΓR+(...

The paper uses a new approach to investigate prime submodules and minimal prime submodules of certain modules such as Artinian and torsion modules. In particular, we introduce a concrete formula for the radical of submodules of Artinian modules.

It is unknown if an Artinian level O-sequence of codimension 3 and type r (≥ 2) is unimodal, while it is known that any Gorenstein O-sequence of codimension 3 is unimodal. We show that some Artinian non-unimodal O-sequence of codimension 3 cannot be level. We also find another non-level case: if some Artinian algebra A of codimension 3 has the Hilbert function H : h0 h1 · · · hd−1 hd · · · hd }...

We observe that the class of left and right artinian left and right morphic rings agrees with the class of artinian principal ideal rings. For R an artinian principal ideal ring and G a group, we characterize when RG is a principal ideal ring; for finite groups G, this characterizes when RG is a left and right morphic ring. This extends work of Passman, Sehgal and Fisher in the case when R is a...