Let $R$ be an integral domain with quotient field $K$ and $f\left(x\right)$ a polynomial of positive degree in $K\left[x\right]$. In this paper we develop a method for studying almost principal uppers to zero ideals. More precisely, we prove that uppers to zero divisorial ideals of the form $I=f\left(x\right)K\left[x\right]\cap R\left[x\right]$ are almost principal in the following two cases: – $J$, the ideal generated by the leading coefficients of $I$, satisfies $J^{-1}=R$. – $I^{-1}$ as the $R\left[x\right]$-submodule of $K\left(x\right)$ is of finite type. Furthermore we prove that for $I=f\left(x\right)K\left[x\right]\cap R\left[x\right]$ we have: – $I^{-1}\cap K\left[x\right]=\left(I{:}_{K\left(x\right)}I\right)$. – If there exists...

An extension $D\subseteq R$ of integral domains is $t$- (resp., $t$-) if ${\left(IR\right)}^{-1}={\left(I^{-1}R\right)}_v$ (resp., ${\left(IR\right)}_v={\left(I_{v}R\right)}_v)$ for every nonzero finitely generated fractional ideal $I$ of $D$. We show that strongly $t$-compatible implies $t$-compatible and give examples to show that the converse does not hold. We also indicate situations where strong $t$-compatibility and its variants show up naturally. In addition, we study integral domains $D$ such that $D\subseteq R$ is strongly $t$-compatible (resp., $t$-compatible) for every overring $R$ of $D$. ...

Let $p$ be a prime number. A finite Galois extension $N/F$ of a number field $F$ with group $G$ has a normal $p$-integral basis ($p$-NIB for short) when ${𝒪}_N^{\prime }$ is free of rank one over the group ring ${𝒪}_F^{\prime }\left[G\right]$. Here, ${𝒪}_F^{\prime }={𝒪}_F[1/p]$ is the ring of $p$-integers of $F$. Let $m=p^e$ be a power of $p$ and $N/F$ a cyclic extension of degree $m$. When ${\zeta }_m\in F^{\times }$, we give a necessary and sufficient condition for $N/F$ to have a $p$-NIB (Theorem 3). When ${\zeta }_m\notin F^{\times }$ and $p\nmid [F\left({\zeta }_m\right):F]$, we show that $N/F$ has a $p$-NIB if and only if $N\left({\zeta }_m\right)/F\left({\zeta }_m\right)$ has a $p$-NIB (Theorem 1). When $p$ divides $[F\left({\zeta }_m\right):F]$, we show that this...

Let ℓ be a rational prime, K be a number field that contains a primitive ℓth root of unity, L an abelian extension of K whose degree over K, [L:K], is divisible by ℓ, a prime ideal of K whose ideal class has order ℓ in the ideal class group of K, and $a_{}$ any generator of the principal ideal ${}^{\ell }$. We will call a prime ideal of K ’reciprocal to ’ if its Frobenius element generates $Gal(K\left(\sqrt[\ell ]{a_{}}\right)/K)$ for every choice of $a_{}$. We then show that becomes principal in L if and only if every reciprocal prime is not...

Let $B$ be a quaternion algebra over a number field $k$. To a pair of Hilbert symbols $\{a,b\}$ and $\{c,d\}$ for $B$ we associate an invariant $\rho ={\rho }_R\left(\left[𝒟\right(a,b\left)\right],\left[𝒟\right(c,d\left)\right]\right)$ in a quotient of the narrow ideal class group of $k$. This invariant arises from the study of finite subgroups of maximal arithmetic kleinian groups. It measures the distance between orders $𝒟(a,b)$ and $𝒟(c,d)$ in $B$ associated to $\{a,b\}$ and $\{c,d\}.$ If $a=c$, we compute ${\rho }_R(\left[𝒟(a,b)\right],\left[𝒟(c,d)\right])$ by means of arithmetic in the field $k\left(\sqrt{a}\right).$ The problem of extending this algorithm to the general case leads to studying a finite...

In this paper, we deal with the study of intermediate domains between a domain $R$ and a domain $T$ such that $T$ is an intersection of localizations of $R$, namely the pair $\left(R,T\right)$. More precisely, we study the pair $\left(R,R_d\right)$ and the pair $\left(R,\tilde{R}\right)$, where $R_d=\cap \left\{R_M\mid M\in \text{Max}\left(R\right),htM=dimR\right\}$ and $\tilde{R}=\cap \left\{R_M\mid M\in \text{Max}\left(R\right),htM\ge 2\right\}$. We prove that, if $R$ is a Jaffard domain, then $\left(R,R_d\left[n\right]\right)$ is a Jaffard pair, which generalize [5, Théorème 1.9]. We also show that if $R$ is an $S$-domain, then $\left(R,\tilde{R}\right)$ is a residually algebraic pair (that is for each intermediate domain $S$ between $R$ and $\tilde{R}$, if $Q$ is a prime...