Let $R$ be a commutative ring and let $M$ be an $R$-module. We define the small intersection graph $G(M)$ of $M$ with all non-small proper submodules of $M$ as vertices and two distinct vertices $N, K$ are adjacent if and only if $Ncap K$ is a non-small submodule of $M$. In this article, we investigate the interplay between the graph-theoretic properties of $G(M)$ and algebraic properties of $M...

Let V be a possibly singular scheme-theoretic complete intersection subscheme of P over an algebraically closed field of characteristic zero. Using a recent result of Fullwood (“On Milnor classes via invariants of singular subschemes”, Journal of Singularities) we develop an algorithm to compute the Chern-Schwartz-MacPherson class and Euler characteristic of V . This algorithm complements exist...

Let A be a commutative Noetherian ring of dimension n (n ≥ 3). Let I be a local complete intersection ideal in A[T ] of height n. Suppose I/I is free A[T ]/I-module of rank n and (A[T ]/I) is torsion inK0(A[T ]). It is proved in this paper that I is a set theoretic complete intersection ideal in A[T ] if one of the following conditions holds: (1) n ≥ 5, odd; (2) n is even, and A contains the fi...

We show that the associated form, or, equivalently, a Macaulay inverse system, of an Artinian complete intersection of type (d, . . . , d) is polystable. As an application, we obtain an invariant-theoretic variant of the Mather-Yau theorem for homogeneous hypersurface singularities.

This paper provides a model theoretic semantics to feature terms augmented with set descriptions. We provide constraints to specify HPSG style set descriptions, fixed cardinality set descriptions, set-membership constraints, restricted universal role quantifications, set union, intersection, subset and disjointness. A sound, complete and terminating consistency checking procedure is provided to...

Using arithmetic conditions on affine semigroups we prove that for a simplicial toric variety of codimension 2 the property of being a set-theoretic complete intersection on binomials in characteristic p holds either for all primes p, or for no prime p, or for exactly one prime p.

‎let $r$ be a commutative ring with non-zero identity. ‎we describe all $c_3$‎- ‎and $c_4$-free intersection graph of non-trivial ideals of $r$ as well as $c_n$-free intersection graph when $r$ is a reduced ring. ‎also, ‎we shall describe all complete, ‎regular and $n$-claw-free intersection graphs. ‎finally, ‎we shall prove that almost all artin rings $r$ have hamiltonian intersection graphs. ...