Related papers: Prime ideals in the Boolean polynomial semiring

In this thesis, we focus on the study of some classes of monomial ideals, namely lexsegment ideals and monomial ideals with linear quotients.

Binomial ideals are special polynomial ideals with many algorithmically and theoretically nice properties. We discuss the problem of deciding if a given polynomial ideal is binomial. While the methods are general, our main motivation and…

We study quotients of the Weyl algebra by left ideals whose generators consist of an arbitrary Z^d-graded binomial ideal I along with Euler operators defined by the grading and a parameter in C^d. We determine the parameters for which these…

We show that every proper, dense ideal in a C*-algebra is contained in a prime ideal. It follows that a subset generates a C*-algebra as a not necessarily closed ideal if and only if it is not contained in any prime ideal. This allows us to…

Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring over a field and $A$ a standard graded $S$-algebra. In terms of the Gr\"obner basis of the defining ideal $J$ of $A$ we give a condition, called the x-condition, which implies that all graded…

We survey theory developed over the past 10 years of semirings which need not be additively cancellative. The main feature is a specified ``null ideal'' $\mcA_0$ of a semiring $\mcA,$ taking the place of a zero element, which permits…

In this paper we study primality and primary decomposition of certain ideals which are generated by homogeneous degree $2$ polynomials and occur naturally from determinantal conditions. Normality is derived from these results.

For certain weak versions of the Axiom of Choice (most notably, the Boolean Prime Ideal theorem), we obtain equivalent formulations in terms of partial orders, and filter-like objects within them intersecting certain dense sets or…

Under ABC, Silverman showed that there are infinitely many non-Wieferich primes with respect to any (non-trivial) base $a$. Recently Srinivas and Subramani proved an analogous result over number fields with trivial class group. In the first…

We define a new type of ideal basis called the proper basis that improves both Gr\"obner basis and Buchberger's algorithm. Let $x_1$ be the least variable of a monomial ordering in a polynomial ring $K[x_1,\dotsc,x_n]$ over a field $K$. The…

In this paper, we consider graded near-rings over a monoid $G$ as a generalizations of graded rings over groups. We introduce certain innovative graded prime ideals and study some of its basic properties over graded near-rings.

Let $R=K[x_1,...,x_n]$ be the polynomial ring in $n$ variables over a field $K$ and $I$ be a monomial ideal generated in degree $d$. Bandari and Herzog conjectured that a monomial ideal $I$ is polymatroidal if and only if all its monomial…

We prove that if the initial ideal of a prime ideal is Borel-fixed and the dimension of the quotient ring is less than or equal to two, then given any non-minimal associated prime ideal of the initial ideal it contains another associated…

Over an arbitrary field $\mathbb{F}$, Harbourne conjectured that $$I^{(N (r-1)+1)} \subseteq I^r$$ for all $r>0$ and all homogeneous ideals $I$ in $S = \mathbb{F} [\mathbb{P}^N] = \mathbb{F} [x_0, \ldots, x_N]$. The conjecture has been…

We examine an elementary problem on prime divisibility of binomial coefficients. Our problem is motivated by several related questions on alternating groups.

We study certain types of ideals in the standard Borel subalgebra of an untwisted affine Lie algebra. We classify these ideals in terms of the root combinatorics and give an explicit formula for the number of such ideals in type $A$. The…

We provide a sufficient condition for a polynomial ring, not necessarily commutative, to have a first-order definition for the rational integers.

We continue the analysis of prime and semiprime operations over one-dimensional domains started in \cite{Va}. We first show that there are no bounded semiprime operations on the set of fractional ideals of a one-dimensional domain. We then…

For a nonempty subset $X$ of a ring $R$, the ring $R$ is called $X$-semiprime if, given $a\in R$, $aXa=0$ implies $a=0$. This provides a proper class of semiprime rings. First, we clarify the relationship between idempotent semiprime and…

A semiring generalises the notion of a ring, replacing the additive abelian group structure with that of a commutative monoid. In this paper, we study a notion positioned between a ring and a semiring -- a semiring whose additive monoid is…