Let B be a commutative $\mathbb {Z}$ -graded domain of characteristic zero. An element f is said to cylindrical if it nonzero, homogeneous nonzero degree, and such that B(f) polynomial ring in one variable over subring. We study the relation between existence locally nilpotent derivation B. Also, given d ≥ 1, we give sufficient conditions guarantee every $B^{(d)} = {\bigoplus }_{i \in \mathbb {...

In this article, we generalize the concept of group-graded modules by introducing G-weak graded R-modules, which are R-modules a set G left coset representatives, where R is ring. Moreover, prove some properties these modules. Finally, results related to fields and their vector spaces deduced. Many considerable examples provided with more emphasis on symmetric group S3 dihedral D6, gives symmet...

An integral domain $D$ is called a emph{locally GCD domain} if $D_{M}$ is aGCD domain for every maximal ideal $M$ of $D$. We study somering-theoretic properties of locally GCD domains. E.g., we show that $%D$ is a locally GCD domain if and only if $aDcap bD$ is locally principalfor all $0neq a,bin D$, and flat overrings of a locally GCD domain arelocally GCD. We also show that the t-class group...