Narrow Results

Let $M$ be a torsion-free module over an integral domain $D$ with quotient field $K$. We define a concept of completely integrally closed modules in order to study Krull modules. It is shown that a Krull module $M$ is a $v$-multiplication module if and only if ${(𝔭M)}_v$ is a maximal $v$-submodule and ${({𝔭}^{-1}𝔭)}_{v_1}=D$ for every minimal prime ideal $𝔭$ of $D$. If $M$ is a finitely generated Krull module, then $M_1=\cap M_{𝔭}$ is a Krull module and $v$-multiplication module. It is also shown that the following three conditions are equivalent: $M$ is completely integrally closed, $M[x]$ is completely integrally closed, and $M[[x]]$ is completely integrally closed.

An additive mappings $\delta$ on $R$ is called a derivation if $\delta (ab)=\delta (a)b+a\delta (b)$ for all $a,b\in R$. If $\delta$ is a derivation on a ring $R$, $M$ and $N$ are right $R$-modules and $f$ is a right $R$-linear mapping from $M$ to $N$, then an additive mapping $d:M\to N$ is called a $(\delta ,f)$-derivation if $d(xa)=d(x)a+f(x)\delta (a)$ for all $x\in M$ and $a\in R$. If $\delta$ is determined, then the $(\delta ,f)$-derivation is written briefly as $f$-derivation. The aim of this paper is to initiate the study of $f$-derivations on polynomial modules. In fact, we induce the derivations of polynomial rings $R[x]$ based on the derivations of rings. In addition, we define $R[x]$-linear mapping on a polynomial module over a polynomial ring based on $R$-linear mapping on modules. Finally, we induce the $f$-derivations on polynomial modules over a polynomial ring based on $f$-derivations of modules and apply this result to the semigroup rings.

Let $\mathfrak{g}$ be a finite dimensional complex simple Lie algebra with Cartan subalgebra $\mathfrak{h}$. Then $\mathbb{C}[\mathfrak{h}]$ has a $\mathfrak{g}$-module structure if and only if $\mathfrak{g}$ is of type $A$ or of type $C$; this is called the polynomial module of rank one. In the quantum version, the rank one polynomial modules over $U_q({\mathfrak{sl}}_2)$ have been classified. In this paper, we prove that the quantum group $U_q({\mathfrak{sl}}_3)$ has no rank one polynomial module.