Rings with S-acc on d-annihilators

A commutative ring [Formula: see text] is said to satisfy acc on d-annihilators if for every sequence [Formula: see text] of elements of [Formula: see text] the sequence [Formula: see text] is stationary. In this paper we extend the notion of rings with acc on d-annihilators by introducing the concept of rings with [Formula: see text]-acc on d-annihilators, where [Formula: see text] is a multiplicative set. Let [Formula: see text] be a commutative ring and [Formula: see text] a multiplicative subset of [Formula: see text] We say that [Formula: see text] satisfies [Formula: see text]-acc on d-annihilators if for every sequence [Formula: see text] of elements of [Formula: see text] the sequence [Formula: see text] is [Formula: see text]-stationary, that is, there exist a positive integer [Formula: see text] and an [Formula: see text] such that for each [Formula: see text] [Formula: see text] We give equivalent conditions for the power series (respectively, polynomial) ring over an Armendariz ring to satisfy [Formula: see text]-acc on d-annihilators. We also study serval properties of rings satisfying [Formula: see text]-acc on d-annihilators. The concept of the amalgamated duplication of [Formula: see text] along an ideal [Formula: see text] [Formula: see text] is studied.

On the regularity of product of irreducible monomial ideals

Let [Formula: see text] be a polynomial ring in [Formula: see text] variables over a field [Formula: see text]. When [Formula: see text], [Formula: see text] and [Formula: see text] are monomial ideals of [Formula: see text] generated by powers of the variables [Formula: see text], it is proved that [Formula: see text]. If [Formula: see text], the same result for the product of a finite number of ideals as above is proved.

Elliptic curve over a finite ring generated by 1 and an idempotent element $e$ with coefficients in the finite field $\mathbb{F}_{3^d}$

An elliptic curve over a ring $\mathcal{R}$ is a curve in the projective plane $\mathbb{P}^{2}(\mathcal{R})$ given by a specific equation of the form $f(X, Y, Z)=0$ named the Weierstrass equation, where $f(X, Y, Z)=Y^2Z+a_1XYZ+a_3YZ^2-X^3-a_2X^2Z-a_4XZ^2-a_6Z^3$ with coefficients $a_1, a_2, a_3, a_4, a_6$ in $\mathcal{R}$ and with an invertible discriminant in the ring $\mathcal{R}.$ %(see \cite[Chapter III, Section 1]{sil1}). In this paper, we consider an elliptic curve over a finite ring of characteristic 3 given by the Weierstrass equation: $Y^2Z=X^3+aX^2Z+bZ^3$ where $a$ and $b$ are in the quotient ring $\mathcal{R}:=\mathbb{F}_{3^d}[X]/(X^2-X),$ where $d$ is a positive integer and $\mathbb{F}_{3^d}[X]$ is the polynomial ring with coefficients in the finite field $\mathbb{F}_{3^d}$ and such that $-a^3b$ is invertible in $\mathcal{R}$.

Multigraded Castelnuovo–Mumford regularity via Klyachko filtrations

In this paper, we consider Z r \mathbb{Z}^{r} -graded modules on the Cl ⁡ ( X ) \operatorname{Cl}(X) -graded Cox ring C ⁢ [ x 1 , … , x r ] \mathbb{C}[x_{1},\dotsc,x_{r}] of a smooth complete toric variety 𝑋. Using the theory of Klyachko filtrations in the reflexive case, we construct a collection of lattice polytopes codifying the multigraded Hilbert function of the module. We apply this approach to reflexive Z s + r + 2 \mathbb{Z}^{s+r+2} -graded modules over any non-standard bigraded polynomial ring C ⁢ [ x 0 , … , x s , y 0 , … , y r ] \mathbb{C}[x_{0},\dotsc,x_{s},\allowbreak y_{0},\dotsc,y_{r}] . In this case, we give sharp bounds for the multigraded regularity index of their multigraded Hilbert function, and a method to compute their Hilbert polynomial.

Irreducible skew polynomials over domains

Let S be a domain and R = S[t; σ, δ] a skew polynomial ring, where σ is an injective endomorphism of S and δ a left σ -derivation. We give criteria for skew polynomials f ∈ R of degree less or equal to four to be irreducible. We apply them to low degree polynomials in quantized Weyl algebras and the quantum planes. We also consider f(t) = tm − a ∈ R.

Differential operators and Higher Specht polynomials

In this note, we study the action of the rational quantum Calogero-Moser system on polynomials rings. This a continuation of our paper [Ibrahim Nonkan 2021 J. Phys.: Conf. Ser. 1730 012129] in which we deal with the polynomial representation of the ring of invariant differential operators. Using the higher Specht polynomials we give a detailed description of the actions of the Weyl algebra associated with the ring of the symmetric polynomial C[x 1,..., xn]Sn on the polynomial ring C[x 1,..., xn ]. In fact, we show that its irreducible submodules over the ring of differential operators invariant under the symmetric group are its submodules generated by higher Specht polynomials over the ring of the symmetric polynomials. We end up studying the polynomial representation of the ring of differential operators invariant under the actions of products of symmetric groups by giving the generators of its simple components, thus we give a differential structure to the higher Specht polynomials.

Differential operators and reection group of type Dn

In this note, we study the actions of rational quantum Olshanetsky-Perelomov systems for finite reflections groups of type D n . we endowed the polynomial ring C[x 1,..., xn ] with a differential structure by using directly the action of the Weyl algebra associated with the ring C[x 1,..., xn ] W of invariant polynomials under the reflections groups W after a localization. Then we study the polynomials representation of the ring of invariant differential operators under the reflections groups. We use the higher Specht polynomials associated with the representation of the reflections group W to exhibit the generators of its simple components.

Differential operators and reection group of type Bn

In this note, we study the polynomial representation of the quantum Olshanetsky-Perelomov system for a finite reflection group W of type Bn. We endowed the polynomial ring C[x 1,..., xn ] with a structure of module over the Weyl algebra associated with the ring C[x 1,..., xn]W of invariant polynomials under a reflections group W of type Bn . Then we study the polynomials representation of the ring of invariant differential operators under the reflections group W. We make use of the theory of representation of groups namely the higher Specht polynomials associated with the reflection group W to yield a decomposition of that structure by providing explicitly the generators of its simple components.