In this article by using G-type domains, we introduce strong G-type domains and locally countable quotient rings (lcqr). Morover, G-type ideals are classified. Finally some relations between prime ideals and G-type ideals in valuation rings have been investigated.

In this article, we introduce and examine the concept of $G$-type rings. A ring $R$ is classified as a $G$-type ring if its total quotient ring, denoted by $Q$, is generated by a countable number of elements over $R$ as an $R$-algebra. Formally, $Q = R[S^{-1}]$, where $S$ is a countable set of regular elements in $R$. We establish that $R$ is $G$-type if and only if there exists a countable set of regular elements, denoted as $S$, such that every prime ideal disjoint from $S$ consists solely of zero-divisors.It is shown thatwhenever a ring $T$ is countably generated over a subring $R$, asan $R$-algebra, and $T$ is strongly algebraic over $R$, then $R$is $G$-type if and only if $T$ is $G$-type.

A transitive subgroup $G\leq S_n$ is called a genus $g$ group if there exist non identity elements $x_1,...,x_r\in G$ satisfying $G=\langle x_1,x_2,...,x_r\rangle$, $\prod_{i=1}^r {x_i}=1$ and $\sum_{i=1}^r ind\, x_i=2(n+g-1)$. The Hurwitz space $\mathcal{H}^{in}_{r,g}(G)$ is the space of genus $g$ covers of the Riemann sphere $\mathbb{P}^1\mathbb{C}$ with $r$ branch points and the monodromy group $G$. Isomorphisms of such covers are in one to one correspondence with genus $g$ groups.In this article, we show that $G$ possesses genus one and two group if it is diagonal type and acts primitively on $\Omega$. Furthermore, we study the connectedness of the Hurwitz space $\mathcal{H}^{in}_{r,g}(G)$ for genus 1 and 2.

Let G be a , nitely generated abelian group and M be a G-graded A-module. In general, G-associated prime ideals to M may not exist. In this paper, we introduce the concept of G-attached prime ideals to M as a generalization of G-associated prime ideals which gives a connection between certain G-prime ideals and G-graded modules over a (not necessarily G-graded Noetherian) ring. We prove that the G-attached prime ideals exist for every nonzero G-graded module and this generalization is proper. We transfer many results of G-associated prime ideals to G-attached prime ideals and give some applications of it.

In this paper, we defined $\alpha_{G}-$admissible interpolative type contraction mappings in $G$-metric spaces. We proved some convergence results for such classes of mappings using the properties of $G$-metric space and found the fixed point results for such contractive mappings. To elaborate on the results we provided some examples, which show that our results hold in the setting of $G-$metric spaces.

We say that an isometric immersion hypersurface x: Mn! En+1 is of null Lk-2-type if x = x1+x2, x1,x2: Mn! En+1 are smooth maps and Lkx1 = 0,Lkx2 = , x2, ,is non-zero real number, Lk is the linearized operator of the (k+1)th mean curvature of the hypersurface, i. e., Lk(f) = tr(Pk, Hessianf) for f 2 C 1 (M), where Pk is the kth New-ton transformation, Lkx = (Lkx1, : : :,Lkxn+1),x = (x1, : : :, xn+1). In this article, we classify , (2)-ideal Euclidean hypersurfaces of null L1-2-type.