Related papers: Symbolic Rees algebras and set-theoretic complete …

For the polynomial ring over an arbitrary field with twelve variables, there exists a prime ideal whose symbolic Rees algebra is not finitely generated.

We apply some basic notions from combinatorial topology to establish various algebraic properties of edge ideals of graphs and more general Stanley-Reisner rings. In this way we provide new short proofs of some theorems from the literature…

Let R be a commutative ring with identity. In this paper, we introduce and investigate the second ideal intersection graph SII(R) of R with vertices are non-zero proper ideals of R and two distinct vertices I and J are adjacent if and only…

Given a 0-dimensional affine K-algebra R=K[x_1,...,x_n]/I, where I is an ideal in a polynomial ring K[x_1,...,x_n] over a field K, or, equivalently, given a 0-dimensional affine scheme, we construct effective algorithms for checking whether…

Recently, nearly complete intersection ideals were defined by Boocher and Seiner to establish lower bounds on Betti numbers for monomial ideals (arXiv:1706.09866). Stone and Miller then characterized nearly complete intersections using the…

A classical result of Micali asserts that a Noetherian local ring is regular if and only if the Rees algebra of its maximal ideal is defined by an ideal of linear forms. In this case, this defining ideal may be realized as a determinantal…

Given two determinantal rings over a field k. We consider the Rees algebra of the diagonal ideal, the kernel of the multiplication map. The special fiber ring of the diagonal ideal is the homogeneous coordinate ring of the join variety.…

When $R$ is a Noetherian ring and we have a family of ideals in which every ideal contains at least one nonzero divisor, then it is already known that the defining ideal of the multi-Rees algebra of these ideals is equal to a saturated…

Given a skew-symmetric matrix $X$, the Pfaffian of $X$ is defined as the square root of the determinant of $X$. In this article, we give the explicit defining equations of the Rees algebra of a Pfaffian ideal $I$ generated by the maximal…

We discuss projective equivalence of ideals in Noetherian rings and the existence or failure of existence of projectively full ideals. We describe connections with the Rees valuations and Rees integers of an ideal, and consider the question…

Let $R$ be a commutative ring and ${\Bbb{A}}(R)$ be the set of ideals with non-zero annihilators. The annihilating-ideal graph of $R$ is defined as the graph ${\Bbb{AG}}(R)$ with vertex set ${\Bbb{A}}(R)^*={\Bbb{A}}\setminus\{(0)\}$ such…

In this paper we define the algebraic sets and the ideal of points for bijective skew PBW extensions with coefficients in left Noetherian domains. Some properties of affine algebraic sets of commutative algebraic geometry will be extended,…

We are interested in the structure of almost complete intersection ideals of grade 3. We give three constructions of these ideals and their free resolutions: one from the commutative algebra point of view, an equivariant construction giving…

Let $E$ be a module of projective dimension one over a Noetherian ring $R$ and consider its Rees algebra $\mathcal{R}(E)$. We study this ring as a quotient of the symmetric algebra $\mathcal{S}(E)$ and consider the ideal $\mathcal{A}$…

In this paper, we define on one hand, the notions of characteristics as well as central characteristics ideals of a given Leibniz algebra g and provide a necessary condition under which for two given subalgebras J and K of g such that, J IS…

In this paper we focus on the structure of the variety of Lie algebras with a finite number of ideals and their graph representations using Hasse diagrams. The large number of necessary conditions on the algebraic structure of this type of…

Let $G=(V,E)$ be a finite, simple graph. We consider for each oriented graph $G_{\cal O}$ associated to an orientation ${\cal O}$ of the edges of $G$, the toric ideal $P_{G_{\cal O}}$. In this paper we study those graphs with the property…

The regularity of the Rees ring of the edge ideal of a finite simple graph is studied. We show that the matching number is a lower and matching number~$+1$ is an upper bound of the regularity, if the Rees algebra is normal. In general the…

Among reduced Noetherian prime characteristic commutative rings, we prove that a regular ring is precisely one where finite intersection of ideals commutes with taking bracket powers. However, reducedness is essential for this equivalence.…

We study the family of graphs whose number of primitive cycles equals its cycle rank. It is shown that this family is precisely the family of ring graphs. Then we study the complete intersection property of toric ideals of bipartite graphs…