Related papers: Binomial canonical decompositions of binomial idea…
We classify all binomial edge ideals that are complete intersection and Cohen-Macaulay almost complete intersection. We also describe an algorithm and provide an implementation to compute primary decomposition of binomial edge ideals.
Taking a ring-theoretic perspective as our motivation, the main aim of this series is to establish a comprehensive theory of ideals in commutative quantales with an identity element. This particular article focuses on an examination of…
We investigate bicomplex analogues of fundamental notions from classical algebraic number theory. In particular, we show that the primitive element theorem admits a natural generalization to bicomplex extensions, giving rise to two distinct…
The fundamental theorem of symmetric polynomials over rings is a classical result which states that every unital commutative ring is fully elementary, i.e. we can express symmetric polynomials with elementary ones in a unique way. The…
We show that every polynomial overring of the ring ${\rm Int}(\mathbb Z)$ of polynomials which are integer-valued over $\mathbb Z$ may be considered as the ring of polynomials which are integer-valued over some subset of $\hat{\mathbb{Z}}$,…
Let R be a nil ring. We prove that primitive ideals in the polynomial ring R[x] in one indeterminate over R are of the form I[x] for some ideals I of R.
In this article, we propose a few sufficient conditions on polynomials having integer coefficients all of whose zeros lie outside a closed disc centered at the origin in the complex plane and deduce the irreducibility over the ring of…
Given a monomial ideal in a polynomial ring over a field, we define the generalized Newton complementary dual of the given ideal. We show good properties of such duals including linear quotients and isomorphisms between the special fiber…
Many classical ring-theoretic results state that an ideal that is maximal with respect to satisfying a special property must be prime. We present a "Prime Ideal Principle" that gives a uniform method of proving such facts, generalizing the…
We describe an algorithm which finds binomials in a given ideal $I\subset\mathbb{Q}[x_1,\dots,x_n]$ and in particular decides whether binomials exist in $I$ at all. Binomials in polynomial ideals can be well hidden. For example, the lowest…
We prove some algebraic results on the ring of matrix differential operators over a differential field in the generality of non-commutative principal ideal rings. These results are used in the theory of non-local Poisson structures.
We study the set of algebraic objects known as vanishing polynomials (the set of polynomials that annihilate all elements of a ring) over general commutative rings with identity. These objects are of special interest due to their close…
We characterize monomial ideals which are intersections of monomial prime ideals and study classes of ideals with this property, among them polymatroidal ideals.
In this article, we disprove a conjecture of F. Alarc\'on and D. Anderson and give a complete classification of the prime ideals in the one variable polynomial semiring with coefficients in Boolean semifield. We group the prime ideals of…
We describe all polarizations for all abelian varieties over a finite field in a fixed isogeny class corresponding to a squarefree Weil polynomial, when one variety in the isogeny class admits a canonical liftings to characteristic zero,…
Can there be a structure space-type theory for an arbitrary class of ideals of a ring? The ideal spaces introduced in this paper allows such a study and our theory includes (but not restricted to) prime, maximal, minimal prime, strongly…
We show that under some conditions, if the initial ideal in$_<(I)$ of an ideal $I$ in a polynomial ring has the property that its symbolic and ordinary powers coincide, then the ideal $I$ shares the same property. We apply this result to…
Recent results of Kahle and Miller give a method of constructing primary decompositions of binomial ideals by first constructing "mesoprimary decompositions" determined by their underlying monoid congruences. Monoid congruences (and…
The geometric and algebraic theory of monomial ideals and multigraded modules is initiated over real-exponent polynomial rings and, more generally, monoid algebras for real polyhedral cones. The main results include the generalization of…
We establish the primary decomposition and uniqueness of primary decomposition for k-ideals in commutative Noetherian semirings.