If M and N are right R-modules, M is called Socle-N-injective (Soc-N-injective) if every R-homomorphism from the socle of N into M extends to N. Equivalently, for every semisimple submodule K of N, any R-homomorphism f : K → M extends to N. In this article, we investigate the notion of soc-injectivity.

One of the most important branches of mathematics is algebraic graph theory, which solves graph problems with algebraic methods. In graph theory, several algebraic properties of a ring can be represented. In this paper, we define an innovative graph on rings, explore its characteristics, and examine how it relates to other notions in the field. Let S be a ring; the quasi‐regular graph of S is a graph with a vertex set of S − {0} and any two different vertices w and z are adjacent if 1 − wz is a unit in S. We study this graph by providing different examples and proving some crucial characteristics. This study provides important results and paves the way for a lot of different inquiries and studies utilizing this novel approach.

Topological indices are numerical invariants derived from graph structures that are essential tools used in computational chemistry and biology for encoding molecular information. By exploiting the inherent symmetries of molecular graphs, we develop efficient algorithms to compute these indices, particularly for large and complex molecules. These indices are rooted in vertex degrees, edge degrees, and other graph parameters, have been extensively studied, and are crucial for understanding the relationship between molecular structure and properties. Recent research has focused on the entire Zagreb indices, which integrate both vertex and edge degrees considering adjacency and incidence relationships. This paper introduces a novel variant, namely, the modified forgotten entire Zagreb index. The efficacy of this new index is underscored by its robust correlation with the physical and chemical properties of octane isomers and lower benzenoid hydrocarbons. Additionally, we derive explicit formulas for this index for several significant graph families.

It is proved here that a ring R is right pseudo-Frobenius if and only if R is a right Kasch ring such that the second right singular ideal is injective.

In this paper, we introduce a new ring of ponderation functions which allowed us to describe and classify a large class of ring of Bessel integral operators. The main result is to give an explicit form of some ideals of such a ring of operators.