<abstract> <p>In this article, we introduce the concepts of graded $ s $-prime submodules which is a generalization of graded prime submodules. We study the behavior of this notion with respect to graded homomorphisms, localization of graded modules, direct product, and idealization. We succeeded to prove the existence of graded $ s $-prime submodules in the case of graded-Noetherian modules. Also, we provide some sufficient conditions for the existence of such objects in the general case, as well as, in the particular case of a grading by a finite group, polycyclic-by-finite group, or by $ \mathbb{Z} $, in addition to the interesting case of crossed product grading, which includes the class of group rings.</p> </abstract>
Let $G$ be a group with identity $e$, $R$ be a $G$-graded ring with unity $1$ and $M$ be a $G$-graded $R$-module. In this article, we introduce the concept of graded quasi multiplication modules, where graded $M$ is said to be graded quasi multiplication if for every graded weakly prime $R$-submodule $N$ of $M$, $N=IM$ for some graded ideal $I$ of $R$. Also, we introduce the concept of graded absorbing multiplication modules; $M$ is said to be graded absorbing multiplication if $M$ has no graded $2$-absorbing $R$-submodules or for every graded $2$-absorbing $R$-submodule $N$ of $M$, $N=IM$ for some graded ideal $I$ of $R$. We prove many results concerning graded weakly prime submodules and graded $2$-absorbing submodules that will be useful in providing several properties of the two classes of graded quasi multiplication modules and graded absorbing multiplication modules.
The Wiener polarity index Wp of a graph is defined as the number of unordered pairs of its vertices at distance 3. The problem of finding trees attaining the maximum Wp value, among all chemical trees of a fixed order n, was solved in the paper (Mol. Inf. 2019, 38, 1800076) for n ≥ 8. Motivated by the usage of Wp in a recent publication (J. Chem. Inf. Model. 2020, 60, 1224–1234), in this article we extend the work done in the aforementioned paper by giving a further ordering of chemical trees with respect to the maximum value of Wp. More precisely, we characterize the trees having the second maximum Wp value (which is 3n − 16) from the class of all chemical trees of a fixed order n, for n ≥ 9.
Let R be a commutative graded ring with unity, S be a multiplicative subset of homogeneous elements of R and P be a graded ideal of R such that P\bigcap S=\emptyset In this article, we introduce several results concerning graded S-prime ideals. Then we introduce the concept of graded weakly S-prime ideals which is a generalization of graded weakly prime ideals. We say that P is a graded weakly S-prime ideal of R if there exists s\in S such that for all x, y\in h(R), if 0\neq xy\in P, then sx\in P or sy\in P. We show that graded weakly S-prime ideals have many acquaintance properties to these of graded weakly prime ideals.
By swapping out atoms for vertices and bonds for edges, a graph may be used to model any molecular structure. A graph G is considered to be a chemical graph in graph theory if no vertex of G has a degree of 5 or greater. The bond incident degree (BID) index for a chemical graph G is defined as the total of contributions f(dG(u),dG(v)) from all edges uv of G, where dG(w) stands for the degree of a vertex w of G, E(G) is the set of edges of G, and f is a real-valued symmetric function. This paper addresses the problem of finding graphs with extremum BID indices over the class of all chemical graphs of a fixed number of edges and vertices.
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