For 0 < γ ≤ 1, let $\Lambda {⁺}_{\gamma }$ be the big Lipschitz algebra of functions analytic on the open unit disc which satisfy a Lipschitz condition of order γ on ̅. For a closed set E on the unit circle and an inner function Q, let $J_{\gamma }(E,Q)$ be the closed ideal in $\Lambda {⁺}_{\gamma }$ consisting of those functions $f\in \Lambda {⁺}_{\gamma }$ for which (i) f = 0 on E, (ii) $|f\left(z\right)-f\left(w\right)|=o\left(\right|z-w{|}^{\gamma })$ as d(z,E),d(w,E) → 0, (iii) $f/Q\in \Lambda {⁺}_{\gamma }$. Also, for a closed ideal I in $\Lambda {⁺}_{\gamma }$, let $E_I$ = z ∈ : f(z) = 0 for every f ∈ I and let $Q_I$ be the greatest common divisor of the inner parts of non-zero functions...

Let $𝔞\subseteq ℂ[x_1,...,x_n]$ be a monomial ideal and $𝒥\left({𝔞}^c\right)$ the multiplier ideal of $𝔞$ with coefficient $c$. Then $𝒥\left({𝔞}^c\right)$ is also a monomial ideal of $ℂ[x_1,...,x_n]$, and the equality $𝒥\left({𝔞}^c\right)=𝔞$ implies that $0<c<n+1$. We mainly discuss the problem when $𝒥\left(𝔞\right)=𝔞$ or $𝒥\left({𝔞}^{n+1-\epsilon }\right)=𝔞$ for all $0<\epsilon <1$. It is proved that if $𝒥\left(𝔞\right)=𝔞$ then $𝔞$ is principal, and if $𝒥\left({𝔞}^{n+1-\epsilon }\right)=𝔞$ holds for all $0<\epsilon <1$ then $𝔞=(x_1,...,x_n)$. One global result is also obtained. Let $\widetilde{𝔞}$ be the ideal sheaf on ${\mathbb{P}}^{n-1}$ associated with $𝔞$. Then it is proved that the equality $𝒥\left(\widetilde{𝔞}\right)=\widetilde{𝔞}$ implies that $\widetilde{𝔞}$ is principal.

Let $R$ be a commutative ring with identity. A proper ideal $I$ is said to be an $n$-ideal of $R$ if for $a,b\in R$, $ab\in I$ and $a\notin \sqrt{0}$ imply $b\in I$. We give a new generalization of the concept of $n$-ideals by defining a proper ideal $I$ of $R$ to be a semi $n$-ideal if whenever $a\in R$ is such that $a^2\in I$, then $a\in \sqrt{0}$ or $a\in I$. We give some examples of semi $n$-ideal and investigate semi $n$-ideals under various contexts of constructions such as direct products, homomorphic images and localizations. We present various characterizations of this new...

Let $G$ be a finite group, and let $N\left(G\right)$ be the set of conjugacy class sizes of $G$. By Thompson’s conjecture, if $L$ is a finite non-abelian simple group, $G$ is a finite group with a trivial center, and $N\left(G\right)=N\left(L\right)$, then $L$ and $G$ are isomorphic. Recently, Chen et al. contributed interestingly to Thompson’s conjecture under a weak condition. They only used the group order and one or two special conjugacy class sizes of simple groups and characterized successfully sporadic simple groups (see Li’s PhD dissertation)....

If $K$ is the splitting field of the polynomial $f\left(x\right)=x^4+p{x}^2+p$ and $p$ is a rational prime of the form $4+n^2$, we give appropriate generators of $K$ to obtain the explicit factorization of the ideal $q{𝒪}_K$, where $q$ is a positive rational prime. For this, we calculate the index of these generators and integral basis of certain prime ideals.

Let $G$ be a finite group and let $N\left(G\right)$ denote the set of conjugacy class sizes of $G$. Thompson’s conjecture states that if $G$ is a centerless group and $S$ is a non-abelian simple group satisfying $N\left(G\right)=N\left(S\right)$, then $G\cong S$. In this paper, we investigate a variation of this conjecture for some symmetric groups under a weaker assumption. In particular, it is shown that $G\cong \mathrm{Sym}(p+1)$ if and only if $\left|G\right|=(p+1)!$ and $G$ has a special conjugacy class of size $(p+1)!/p$, where $p>5$ is a prime number. Consequently, if $G$ is a centerless group with $N\left(G\right)=N\left(\mathrm{Sym}\right(p+1\left)\right)$, then...

Let $H$ be a finite abelian group of odd order, $𝒟$ be its generalized dihedral group, i.e., the semidirect product of $C_2$ acting on $H$ by inverting elements, where $C_2$ is the cyclic group of order two. Let $\Omega \left(𝒟\right)$ be the Burnside ring of $𝒟$, $\Delta \left(𝒟\right)$ be the augmentation ideal of $\Omega \left(𝒟\right)$. Denote by ${\Delta }^n\left(𝒟\right)$ and $Q_n\left(𝒟\right)$ the $n$th power of $\Delta \left(𝒟\right)$ and the $n$th consecutive quotient group ${\Delta }^n\left(𝒟\right)/{\Delta }^{n+1}\left(𝒟\right)$, respectively. This paper provides an explicit $\mathbb{Z}$-basis for ${\Delta }^n\left(𝒟\right)$ and determines the isomorphism class of $Q_n\left(𝒟\right)$ for each positive integer $n$.

For a finite group $G$ denote by $N\left(G\right)$ the set of conjugacy class sizes of $G$. In 1980s, J. G. Thompson posed the following conjecture: If $L$ is a finite nonabelian simple group, $G$ is a finite group with trivial center and $N\left(G\right)=N\left(L\right)$, then $G\cong L$. We prove this conjecture for an infinite class of simple groups. Let $p$ be an odd prime. We show that every finite group $G$ with the property $Z\left(G\right)=1$ and $N\left(G\right)=N\left(A_i\right)$ is necessarily isomorphic to $A_i$, where $i\in \{2p,2p+1\}$.

Let $I$ be an ideal in a commutative Noetherian ring $R$. Then the ideal $I$ has the strong persistence property if and only if $\left(I^{k+1}{:}_{R}I\right)=I^k$ for all $k$, and $I$ has the symbolic strong persistence property if and only if $\left(I^{(k+1)}{:}_R{I}^{\left(1\right)}\right)=I^{\left(k\right)}$ for all $k$, where $I^{\left(k\right)}$ denotes the $k$th symbolic power of $I$. We study the strong persistence property for some classes of monomial ideals. In particular, we present a family of primary monomial ideals failing the strong persistence property. Finally, we show that every square-free monomial...

We show a $p$-parity result in a $D_{2p^n}$-extension of number fields $L/K$ ($p\ge 5$) for the twist $1\oplus \eta \oplus \tau$: $W(E/K,1\oplus \eta \oplus \tau )={(-1)}^{〈1\oplus \eta \oplus \tau ,X_p(E/L)〉}$, where $E$ is an elliptic curve over $K$, $\eta$ and $\tau$ are respectively the quadratic character and an irreductible representation of degree $2$ of $\mathrm{Gal}(L/K)=D_{2p^n}$, and $X_p(E/L)$ is the $p$-Selmer group. The main novelty is that we use a congruence result between ${\epsilon }_0$-factors (due to Deligne) for the determination of local root numbers in bad cases (places of additive reduction above 2 and 3). We also give applications to the $p$-parity conjecture...

We prove that Bloch’s conjecture is true for surfaces with $p_g=0$ obtained as $0$-sets $X_{\sigma }$ of a section $\sigma$ of a very ample vector bundle on a variety $X$ with “trivial” Chow groups. We get a similar result in presence of a finite group action, showing that if a projector of the group acts as $0$ on holomorphic $2$-forms of $X_{\sigma }$, then it acts as $0$ on $0$-cycles of degree $0$ of $X_{\sigma }$. In higher dimension, we also prove a similar but conditional result showing that the generalized Hodge conjecture for general $X_{\sigma }$...