Volume 35, Issue 5

A subset S of an integral domain R is called a semidomain provided that the pairs ( S , + ) {(S,+)} and ( S , ⋅ ) {(S,\cdot\,)} are semigroups with identities. The study of factorizations in integral domains was initiated by Anderson, Anderson, and Zafrullah in 1990, and this area has been systematically investigated since then. In this paper, we study the divisibility and arithmetic of factorizations in the more general context of semidomains. We are specially concerned with the ascent of the most standard divisibility and factorization properties from a semidomain to its semidomain of (Laurent) polynomials. As in the case of integral domains, here we prove that the properties of satisfying the ascending chain condition on principal ideals, having bounded factorizations, and having finite factorizations ascend in the class of semidomains. We also consider the ascent of the property of being atomic and that of having unique factorization (none of them ascends in general). Throughout the paper, we provide several examples aiming to shed some light upon the arithmetic of factorizations of semidomains.

A two-step nilpotent Leibniz superalgebra L is called extra-special if the center and the derived subalgebra of L are equal and both are of dimension one. In this paper, the structure, the capability and the second homology of extra-special Leibniz superalgebras are determined.

A restricted artinian ring is a commutative ring with an identity in which every proper homomorphic image is artinian. Cohen proved that a commutative ring R is restricted artinian if and only if it is noetherian and every nonzero prime ideal of R is maximal. Facchini and Nazemian called a commutative ring isoartinian if every descending chain of ideals becomes stationary up to isomorphism. We show that every proper homomorphic image of a commutative noetherian ring R is isoartinian if and only if R has one of the following forms: (a) R is a noetherian domain of Krull dimension one which is not a principal ideal domain; (b) R ≅ D 1 × ⋯ × D k × A 1 × ⋯ × A l {R\cong D_{1}\times\cdots\times D_{k}\times A_{1}\times\cdots\times A_{l}} , where each D i {D_{i}} is a principal ideal domain and each A i {A_{i}} is an artinian local ring (either k or l may be zero); (c) R is a noetherian ring of Krull dimension one, simple unique minimal prime ideal 𝔭 {\mathfrak{p}} , and R / 𝔭 {R/\mathfrak{p}} is a principal ideal domain. As an application of our result, we describe commutative rings whose proper homomorphic images are principal ideal rings. Some relevant examples are provided.

We improve on the spectral large sieve inequality for symmetric-squares. We also prove a lower bound showing that the most optimistic upper bound is not true for this family.

Let 𝔭 {\mathfrak{p}} be a parabolic subalgebra of 𝔰 ⁢ 𝔩 ⁢ ( V ) {\mathfrak{sl}(V)} of maximal dimension and let 𝔫 ⊂ 𝔭 {\mathfrak{n}\subset\mathfrak{p}} be the corresponding nilradical. In this paper, we classify the set of 𝔰 ⁢ 𝔩 ⁢ ( V ) {\mathfrak{sl}(V)} -modules whose restriction to U ⁢ ( 𝔫 ) {U(\mathfrak{n})} is free of rank 1. It turns out that isomorphism classes of such modules are parametrized by polynomials in dim ⁡ V - 1 {\dim V-1} variables. We determine the submodule structure for these modules and we show that they generically are simple.

Let G be a group, let A = ⊕ g ∈ G A g {A=\bigoplus_{g\in G}A_{g}} be an epsilon-strongly graded ring over G , let R := A 1 {R:=A_{1}} be the homogeneous component associated with the identity of G , and let 𝙿𝚒𝚌𝚂 ⁢ ( R ) {\mathtt{PicS}(R)} be the Picard semigroup of R . In the first part of this paper, we prove that the isomorphism class [ A g ] {[A_{g}]} is an element of 𝙿𝚒𝚌𝚂 ⁢ ( R ) {\mathtt{PicS}(R)} for all g ∈ G {g\in G} . Moreover, the association g ↦ [ A g ] {g\mapsto[A_{g}]} determines a partial representation of G on 𝙿𝚒𝚌𝚂 ⁢ ( R ) {\mathtt{PicS}(R)} which induces a partial action γ of G on the center Z ⁢ ( R ) {Z(R)} of R . Sufficient conditions for A to be an Azumaya R γ {R^{\gamma}} -algebra are presented if R is commutative. In the second part, we study when B is a partial crossed product in the following cases: B = M n ⁡ ( A ) {B=\operatorname{M}_{n}(A)} is the ring of matrices with entries in A , or B = END A ( M ) = ⊕ l ∈ G Mor A ( M , M ) l {B=\operatorname{END}_{A}(M)=\bigoplus_{l\in G}\operatorname{Mor}_{A}(M,M)_{l}} is the direct sum of graded endomorphisms of graded left A -modules M with degree l , or B = END A ⁡ ( M ) {B=\operatorname{END}_{A}(M)} where M = A ⊗ R N {M=A\otimes_{R}N} is the induced module of a left R -module N . Assuming that R is semiperfect, we prove that there exists a subring of A which is an epsilon-strongly graded ring over a subgroup of G and it is graded equivalent to a partial crossed product.

In this paper, a class of non-weight modules over N = 1 {N=1} Lie superalgebras of Block type (without center) is studied. These modules over Ramond algebras of Block type are free of rank 1 when restricted to the universal enveloping algebra of the Cartan subalgebra, while such modules are free of rank 2 over the Neveu–Schwarz algebras of Block type. We present the classification of these modules. Furthermore, the simplicity of these modules is determined.

H. L. Skala (1992) gave the first elegant first-order axiom system for hyperbolic geometry by replacing Menger’s axiom involving projectivities with the theorems of Pappus and Desargues for the hyperbolic plane. In so doing, Skala showed that hyperbolic geometry is incidence geometry. We improve upon Skala’s formulation by doing away with Pappus and Desargues altogether, by substituting for them two simpler axioms.

In this paper, we introduce the mixed radial-angular central Campanato space ℭ ⁢ L rad p 2 , λ ⁢ L ang p 1 ⁢ ( ℝ n ) {{\mathfrak{C}L^{p_{2},\lambda}_{\mathrm{rad}}L^{p_{1}}_{\mathrm{ang}}(\mathbb% {R}^{n})}} and the mixed radial-angular central Morrey space 𝔐 ⁢ L rad p 2 , λ ⁢ L ang p 1 ⁢ ( ℝ n ) {{\mathfrak{M}L^{p_{2},\lambda}_{\mathrm{rad}}L^{p_{1}}_{\mathrm{ang}}(\mathbb% {R}^{n})}} , respectively. Characterizations of the mixed radial-angular central Campanato space are established via the boundedness of the commutators of Hardy type on the mixed radial-angular central Morrey space. Moreover, some characterizations of the mixed radial-angular λ-central bounded mean oscillation space CMO ⁡ L rad p 2 , λ ⁢ L ang p 1 ⁢ ( ℝ n ) {{\operatorname{CMO}L^{p_{2},\lambda}_{\mathrm{rad}}L^{p_{1}}_{\mathrm{ang}}(% \mathbb{R}^{n})}} are also obtained.

For a positive integer q ≢ 2 ⁢ ( mod ⁢ 4 ) {q\not\equiv 2~{}(\mathrm{mod}~{}4)} , this work considers the fourth moment of Dirichlet L -functions averaged over both t ∈ [ 0 , T ] {t\in[0,T]} and primitive characters modulo q . An asymptotic formula with a power saving from both the q -aspect and the t -aspect in the error term is obtained.

We consider the Kudla–Millson theta series associated to a quadratic space of signature ( N , N ) {(N,N)} . By combining a “see-saw” argument with the Siegel–Weil formula, we show that its (regularized) integral along a torus attached to a totally real field of degree N is the diagonal restriction of an Eisenstein series. It allows us to express the Fourier coefficients of the diagonal restriction as intersection numbers, which generalizes a result of Darmon, Pozzi and Vonk to totally real fields.

We prove Gibbons’ conjecture for the quasilinear elliptic equation - Δ p ⁢ u + a ⁢ ( u ) ⁢ | ∇ ⁡ u | q = f ⁢ ( u )   in  ⁢ ℝ N , -\Delta_{p}u+a(u)|\nabla u|^{q}=f(u)\quad\text{in }\mathbb{R}^{N}, where N ≥ 2 {N\geq 2} , 2 ⁢ N + 2 N + 2 < p < 2 {\frac{2N+2}{N+2}<p<2} , q ≥ 1 {q\geq 1} and a and f are Lipschitz continuous functions which satisfy some relevant conditions. This conjecture states that every weak solution u ∈ C 1 ⁢ ( ℝ N ) {u\in C^{1}(\mathbb{R}^{N})} of the equation with | u | ≤ 1 {|u|\leq 1} and lim x N → ± ∞ ⁡ u ⁢ ( x ′ , x N ) = ± 1 {\lim_{x_{N}\to\pm\infty}u(x^{\prime},x_{N})=\pm 1} , uniformly in x ′ ∈ ℝ N - 1 {x^{\prime}\in\mathbb{R}^{N-1}} , must depend only on x N {x_{N}} and ∂ ⁡ u ∂ ⁡ x N > 0 {\frac{\partial u}{\partial x_{N}}>0} in ℝ N {\mathbb{R}^{N}} . In particular, our result holds for a being non-decreasing on [ - 1 , - 1 + δ ] {[-1,-1+\delta]} and on [ 1 - δ , 1 ] {[1-\delta,1]} and f ⁢ ( u ) = | u | r ⁢ u ⁢ | 1 - u 2 | s ⁢ ( 1 - u 2 ) {f(u)=|u|^{r}u|1-u^{2}|^{s}(1-u^{2})} , where r , s , δ ≥ 0 {r,s,\delta\geq 0} . The main tool we use is an adaptation of the sliding method to the corresponding quasilinear elliptic operator.

Let G = GL ⁡ ( m | n ) {G=\operatorname{GL}(m|n)} be a general linear supergroup over an algebraically closed field 𝐤 {\mathbf{k}} of odd characteristic p . In this paper, we construct Jantzen filtration of Weyl modules V ⁢ ( λ ) {V(\lambda)} for G when λ is a typical weight in the sense of Kac’s definition, and consequently obtain a sum formula for their characters. By Steinberg’s tensor product theorem, it is enough for us to study typical weights with aim to formulate irreducible characters. As an application, it turns out that an irreducible G -module L ⁢ ( λ ) {L(\lambda)} can be realized as a Kac module if and only if λ is p -typical.