The Ring Homomorphisms of Matrix Rings over Skew Generalized Power Series Rings

Let M_n (R_1 [[S_1,≤_1,ω_1]]) and M_n (R_2 [[S_2,≤_2,ω_2]]) be a matrix rings over skew generalized power series rings, where R_1,R_2 are commutative rings with an identity element, (S_1,≤_1 ),(S_2,≤_2 ) are strictly ordered monoids, ω_1:S_1→End(R_1 ),〖 ω〗_2:S_2→End(R_2 ) are monoid homomorphisms. In this research, a mapping τ from M_n (R_1 [[S_1,≤_1,ω_1]]) to M_n (R_2 [[S_2,≤_2,ω_2]]) is defined by using a strictly ordered monoid homomorphism δ:(S_1,≤_1 )→(S_2,≤_2 ), and ring homomorphisms μ:R_1→R_2 and σ:R_1 [[S_1,≤_1,ω_1]]→R_2 [[S_2,≤_2,ω_2]]. Furthermore, it is proved that τ is a ring homomorphism, and also the sufficient conditions for τ to be a monomorphism, epimorphism, and isomorphism are given.

Some Topological and Polynomial Indices (Hosoya and Schultz) for the Intersection Graph of the Subgroup of〖 Z〗_(r^n )

Let be any group with identity element (e) . A subgroup intersection graph of a subset is the Graph with V ( ) = - e and two separate peaks c and d contiguous for c and d if and only if , Where is a Periodic subset of resulting from . We find some topological indicators in this paper and Multi-border (Hosoya and Schultz) of , where , is aprime number.

Pairs of domains where all intermediate domains satisfy S-ACCP

The rings considered in this paper are commutative with identity. If [Formula: see text] is a subring of a ring [Formula: see text], then we assume that [Formula: see text] contains the identity element of [Formula: see text]. Let [Formula: see text] be a multiplicatively closed subset (m.c. subset) of a ring [Formula: see text]. An increasing sequence of ideals [Formula: see text] of [Formula: see text] is said to be [Formula: see text]-stationary if there exist [Formula: see text] and [Formula: see text] such that [Formula: see text] for all [Formula: see text]. This paper is motivated by the research work [A. Hamed and H. Kim, On integral domains in which every ascending chain on principal ideals is [Formula: see text]-stationary, Bull. Korean Math. Soc. 57(5) (2020) 1215–1229]. Let [Formula: see text] be a m.c. subset of an integral domain [Formula: see text]. We say that [Formula: see text] satisfies [Formula: see text]-ACCP if every increasing sequence of principal ideals of [Formula: see text] is [Formula: see text]-stationary. Let [Formula: see text] be a subring of an integral domain [Formula: see text] and let [Formula: see text] be a m.c. subset of [Formula: see text]. We say that [Formula: see text] is an [Formula: see text]-ACCP pair if [Formula: see text] satisfies [Formula: see text]-ACCP for every subring [Formula: see text] of [Formula: see text] with [Formula: see text]. The aim of this paper is to provide some pairs of domains [Formula: see text] such that [Formula: see text] is an [Formula: see text]-ACCP pair, where [Formula: see text] is a m.c. subset of [Formula: see text].

The Evolutionary Fate of Mitochondrial Aminoacyl-tRNA Synthetases in Amitochondrial Organisms

During the endosymbiotic evolution of mitochondria, the genes for aminoacyl-tRNA synthetases were transferred to the ancestral nucleus. A further reduction of mitochondrial function resulted in mitochondrion-related organisms (MRO) with a loss of the organelle genome. The fate of the now redundant ancestral mitochondrial aminoacyl-tRNA synthetase genes is uncertain. The derived protein sequence for arginyl-tRNA synthetase from thirty mitosomal organisms have been classified as originating from the ancestral nuclear or mitochondrial gene and compared to the identity element at position 20 of the cognate tRNA that distinguishes the two enzyme forms. The evolutionary choice between loss and retention of the ancestral mitochondrial gene for arginyl-tRNA synthetase reflects the coevolution of arginyl-tRNA synthetase and tRNA identity elements.

Relations on topologized groups

: In our present paper, topological groups are being discussed, where the relations with counter examples built the interest in the generalized structure. Some of these structures have also been converted into the other structures using topological isomorphism. In our work, the identity element plays the important role in lieu of arbitrary element. The role of topology has the more interest in our discipline.

Identity Graph of Finite Cyclic Groups

This paper discusses how to express a finite group as a graph, specifically about the identity graph of a cyclic group. The term chosen for the graph is an identity graph, because it is the identity element of the group that holds the key in forming the identity graph. Through the identity graph, it can be seen which elements are inverse of themselves and other properties of the group. We will look for the characteristics of identity graph of the finite cyclic group, for both cases of odd and even order.

On the structure of finite groups with dominatable enhanced power graph

Let [Formula: see text] be a group. The enhanced power graph of [Formula: see text] is a graph with vertex set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if there exists [Formula: see text] such that [Formula: see text] and [Formula: see text] for some [Formula: see text]. Also, a vertex of a graph is called dominating vertex if it is adjacent to every other vertex of the vertex set. Moreover, an enhanced power graph is said to be a dominatable graph if it has a dominating vertex other than the identity element. In an article of 2018, Bera and his coauthor characterized all abelian finite groups and nonabelian finite [Formula: see text]-groups such that their enhanced power graphs are dominatable (see [2]). In addition as an open problem, they suggested characterizing all finite nonabelian groups such that their enhanced power graphs are dominatable. In this paper, we try to answer their question. We prove that the enhanced power graph of finite group [Formula: see text] is dominatable if and only if there is a prime number [Formula: see text] such that [Formula: see text] and the Sylow [Formula: see text]-subgroups of [Formula: see text] are isomorphic to either a cyclic group or a generalized quaternion group.

Surjective Identifications of Convex Noetherian Separations in Topological (C, R) Space

The interplay of symmetry of algebraic structures in a space and the corresponding topological properties of the space provides interesting insights. This paper proposes the formation of a predicate evaluated P-separation of the subspace of a topological (C, R) space, where the P-separations form countable and finite number of connected components. The Noetherian P-separated subspaces within the respective components admit triangulated planar convexes. The vertices of triangulated planar convexes in the topological (C, R) space are not in the interior of the Noetherian P-separated open subspaces. However, the P-separation points are interior to the respective locally dense planar triangulated convexes. The Noetherian P-separated subspaces are surjectively identified in another topological (C, R) space maintaining the corresponding local homeomorphism. The surjective identification of two triangulated planar convexes generates a quasiloop–quasigroupoid hybrid algebraic variety. However, the prime order of the two surjectively identified triangulated convexes allows the formation of a cyclic group structure in a countable discrete set under bijection. The surjectively identified topological subspace containing the quasiloop–quasigroupoid hybrid admits linear translation operation, where the right-identity element of the quasiloop–quasigroupoid hybrid structure preserves the symmetry of distribution of other elements. Interestingly, the vertices of a triangulated planar convex form the oriented multiplicative group structures. The surjectively identified planar triangulated convexes in a locally homeomorphic subspace maintain path-connection, where the right-identity element of the quasiloop–quasigroupoid hybrid behaves as a point of separation. Surjectively identified topological subspaces admitting multiple triangulated planar convexes preserve an alternative form of topological chained intersection property.

THE TATTOO AS ANCESTRAL LEGACY AND DICHOTOMIC ELEMENT OF NATIONAL IDENTITY

Nowadays, tattoos are associated mostly with fashion; nevertheless, tattooing can constitute a tool of national identity due to its ancestral relation with the Latin American native culture.With regard to the art of the state, no work can be found that analyzes this specific topic. This paper discusses the determinant footprint of ancient symbols in the culture and its actual use as a national identity element. The study uses a deductive method and qualitative analysis to clear the situated premises and ascertain the problem solution through the use of ancient icons in tattoos as identity symbols, which may be reconstructed from the consideration that supports the national proud and development. Although the results still present a low percentage of users, there are citizens who don tattoos. The concept about its representation is positive, and the use of tattoo as an element of identity was found among 50% of citizens. This leads us to conclude that it is, indeed, used in this sense and that the trend could progressively increase. In order to validate this premise, a quarter of the population was necessary.