On artinian $QF-3$,$1$-Gorenstein rings

Artinian Band Sums of Rings

Band sums of associative rings were introduced by Weissglass in 1973. The main theorem claims that the support of every Artinian band sum of rings is finite. This result is analogous to the well-known theorem on Artinian semigroup rings. 1991 Mathematics subject classification (Amer. Math. Soc): primary 16P20, 16W50; secondary 20M25. Let B be a band, that is, a semigroup consisting of idempoten...

Artinian and Noetherian Rings

On the ubiquity of Gorenstein rings

Introduction APIARY [23~, and subsequently GORENSTEIN Eg] and SAMUEL [2(r proved 9 3-~ t that if (2 is a point on a plane curve then no=2d Q, where dQ=dlmw0/ 9 Q and nQ = dim C'Q/~Q, 9 being the local ring of Q, CQ its normalization, and ~Q the conductor. This condition has received attention from a variety of algebraic geometers, and recently ROQUETTE [19J and BEI~GER [52 have put it i n a rat...

Gorenstein rings through face rings of manifolds

The face ring of a homology manifold (without boundary) modulo a generic system of parameters is studied. Its socle is computed and it is verified that a particular quotient of this ring is Gorenstein. This fact is used to prove that the sphere g-conjecture implies all enumerative consequences of its far reaching generalization (due to Kalai) to manifolds. A special case of Kalai’s manifold g-c...

The central simple modules of Artinian Gorenstein algebras

Let A be a standard graded Artinian K-algebra, with char K = 0. We prove the following. 1. A has the Weak Lefschetz Property (resp. Strong Lefschetz Property) if and only if Gr(z)(A) has the Weak Lefschetz Property (resp. Strong Lefschetz Property) for some linear form z of A. 2. If A is Gorenstein, then A has the Strong Lefschetz Property if and only if there exists a linear form z of A such t...