Mathematics

The theory of τ -factorizations on integral domains was developed by Anderson and Frazier. This theory characterized all the known factorizations and opened the opportunity to create new ones. It can be visualized as a restriction to the structure's multiplicative operation, by considering a symmetric relation τ on the set of non-zero non-unit elements of an integral domain. The main goal of this work is to study the τ -factorization concept, when τ is a composition of two or more relations. To achieve this, the specific properties one can obtain from the given relations are verified and analyzed. Some of the studied properties which are the most known include: reflexivity, symmetry, transitivity, antisymmetry. And others related to the τ -factorization theory, like: divisive, associate-preserving and multiplicative relations.

Generalized Frobenius powers of an ideal were introduced in work of Hernández, Teixeira, and Witt as characteristic-dependent analogs of test ideals. However, little is known about the Frobenius powers and critical exponents of specific ideals, even in the monomial case. We describe an algorithm to compute the critical exponents of monomial ideals and use this algorithm to prove some results about their Frobenius powers and critical exponents. Rather than using test ideals, our algorithm uses techniques from linear optimization.

We present a new algorithm for computing the real radical of an ideal and, more generally, the-radical of , which is based on convex moment optimization. A truncated positive generic linear functional vanishing on the generators of is computed solving a Moment Optimization Problem (MOP). We show that, for a large enough degree of truncation, the annihilator of generates the real radical of. We give an e ective, general stopping criterion on the degree to detect when the prime ideals lying over the annihilator are real and compute the real radical as the intersection of real prime ideals lying over. The method involves several ingredients, that exploit the properties of generic positive moment sequences. A new e cient algorithm is proposed to compute a graded basis of the annihilator of a truncated positive linear functional. We propose a new algorithm to check that an irreducible decomposition of an algebraic variety is real, using a generic real projection to reduce to the hypersurface case. There we apply the Sign Changing Criterion, e ectively performed with an exact MOP. Finally we illustrate our approach in some examples.

It is shown that if R is a ring, p a prime element of an integral domain D≤R with ⋂ ∞ n=1 p n D=0 and p∈U(R) , then R has a conch maximal subring (see \cite{faith}). We prove that either a ring R has a conch maximal subring or U(S)=S∩U(R) for each subring S of R (i.e., each subring of R is closed with respect to taking inverse, see \cite{invsub}). In particular, either R has a conch maximal subring or U(R) is integral over the prime subring of R . We observe that if R is an integral domain with |R|= 2 2 ℵ 0 , then either R has a maximal subring or |Max(R)|= 2 ℵ 0 , and in particular if in addition dim(R)=1 , then R has a maximal subring. If R⊆T be an integral ring extension, Q∈Spec(T) , P:=Q∩R , then we prove that whenever R has a conch maximal subring S with (S:R)=P , then T has a conch maximal subring V such that (V:T)=Q and V∩R=S . It is shown that if K is an algebraically closed field which is not algebraic over its prime subring and R is affine ring over K , then for each prime ideal P of R with ht(P)≥dim(R)−1 , there exists a maximal subring S of R with (S:R)=P . If R is a normal affine integral domain over a field K , then we prove that R is an integrally closed maximal subring of a ring T if and only if dim(R)=1 and in particular in this case (R:T)=0 .

We study the connection between probability distributions satisfying certain conditional independence (CI) constraints, and point and line arrangements in incidence geometry. To a family of CI statements, we associate a polynomial ideal whose algebraic invariants are encoded in a hypergraph. The primary decompositions of these ideals give a characterisation of the distributions satisfying the original CI statements. Classically, these ideals are generated by 2-minors of a matrix of variables, however, in the presence of hidden variables, they contain higher degree minors. This leads to the study of the structure of determinantal hypergraph ideals whose decompositions can be understood in terms of point and line configurations in the projective space.

The first goal of the present paper is to study the class groups of the edge rings of complete multipartite graphs, denoted by k[ K r 1 ,…, r n ] , where 1≤ r 1 ≤⋯≤ r n . More concretely, we prove that the class group of k[ K r 1 ,…, r n ] is isomorphic to Z n if n=3 with r 1 ≥2 or n≥4 , while it turns out that the excluded cases can be deduced into Hibi rings. The second goal is to investigate the special class of divisorial ideals of k[ K r 1 ,…, r n ] , called conic divisorial ideals. We describe conic divisorial ideals for certain K r 1 ,…, r n including all cases where k[ K r 1 ,…, r n ] is Gorenstein. Finally, we give a non-commutative crepant resolution (NCCR) of k[ K r 1 ,…, r n ] in the case where it is Gorenstein.

Let I⊆S=K[ x 1 ,…, x n ] be a homogeneous ideal equipped with a monomial order < . We show that if in < (I) is a square-free monomial ideal, then S/I and S/ in < (I) have the same connectedness dimension. We also show that graphs related to connectedness of these quotient rings have the same number of components. We also provide consequences regarding Lyubeznik numbers. We obtain these results by furthering the study of connectedness modulo a parameter in a local ring.

We study several consequences of the packing problem, a conjecture from combinatorial optimization, using algebraic invariants of square-free monomial ideals. While the packing problem is currently unresolved, we successfully settle the validity of its consequences. Our work prompts additional questions and conjectures, which are presented together with their motivation.

A local ring R is regular if and only if every finitely generated R -module has finite projective dimension. Moreover, the residue field k is a test module: R is regular if and only if k has finite projective dimension. This characterization can be extended to the bounded derived category D f (R) , which contains only small objects if and only if R is regular. Recent results of Pollitz, completing work initiated by Dwyer-Greenlees-Iyengar, yield an analogous characterization for complete intersections: R is a complete intersection if and only if every object in D f (R) is proxy small. In this paper, we study a return to the world of R -modules, and search for finitely generated R -modules that are not proxy small whenever R is not a complete intersection. We give an algorithm to construct such modules in certain settings, including over equipresented rings and Stanley-Reisner rings.

The notion of strict closedness of rings was given by J. Lipman in connection with a conjecture of O. Zariski. The present purpose is to give a practical method of construction of strictly closed rings. It is also shown that the Stanley-Reisner rings of simplicial complexes (resp. F -pure rings satisfying the condition ( S 2 ) of Serre) are strictly closed (resp. weakly Arf) rings.