Algebraic structures and their applications
Year:2024 | Volume:11 | Issue:1|Papers Count:12

An $R-$module $M$ is called almost uniserial if any two non-isomorphic submodules of $M$ are comparable. A ring $R$ is an almost left uniserial ring if $_R R$ is almost uniserial. In this paper, we introduce a class of artinian almost uniserial rings. Also we give a classification of almost uniserial modules over principal ideal domains.

The notions of a local automorphism for Lie algebras are defined as similar to the associative case. Every automorphism of a Lie algebra $\mathcal{L}$ is a local automorphism. For a given Lie algebra $\mathcal{L}$, the main problem concerning these notions is to prove that they automatically become an automorphism or to give examples of local automorphisms of $\mathcal{L}$, which are not automorphisms. In this paper, we study local automorphisms on quasi-filiform Leibniz algebras. It is proved that quasi-filiform Leibniz algebras of type I, as a rule, admit local automorphisms which are not automorphisms.

Let $R$ be a commutative ring with unity and $A(R)$ be the set of annihilating-ideals of $R$. The annihilator intersection graph of $R$, represented by $AIG(R)$, is an undirected graph with $A(R)^*$ as the vertex set and $\mathfrak{M} \sim \mathfrak{N}$ is an edge of $AIG(R)$ if and only if $Ann(\mathfrak{M}\mathfrak{N}) \neq Ann(\mathfrak{M}) \cap Ann(\mathfrak{N})$, for distinct vertices $\mathfrak{M}$ and $\mathfrak{N}$ of $AIG(R)$. In this paper, we first defined finite commutative rings whose annihilator intersection graph is isomorphic to various well-known graphs, and then all finite commutative rings with a planar or toroidal annihilator intersection graph were characterized.

The fuzzy set is a fantastic tool for expressing hesitancy and dealing with uncertainty in real-world circumstances. Soft set theory has recently been developed to deal with practical problems. The soft and fuzzy sets were combined by Jun et al. to generate hybrid structures. The idea of hybrid ideals on a distributive lattice is discussed in this work. The relation between hybrid congruences and hybrid ideals on a distributive lattice is also examined. In addition, the product of hybrid ideals and its numerous results are discussed.

A ring $R$ is called radically $z$-covered (resp. radically $z^\circ$-covered) if every $\sqrt z$-ideal (resp. $\sqrt {z^\circ}$-ideal) in $R$ is a higher order $z$-ideal (resp. $z^\circ$-ideal). In this article we show with a counter-example that a ring may not be radically $z$-covered (resp. radically $z^\circ$-covered). Also a ring $R$ is called $z^\circ$-terminating if there is a positive integer $n$ such that for every $m\geq n$, each $z^{\circ m}$-ideal is a $z^{\circ n}$-ideal. We show with a counter-example that a ring may not be $z^\circ$-terminating. It is well known that whenever a ring homomorphism $\phi:R\to S$ is strong (meaning that it is surjective and for every minimal prime ideal $P$ of $R$, there is a minimal prime ideal $Q$ of $S$ such that $\phi^{-1}[Q] = P$), and if $R$ is a $z^\circ$-terminating ring or radically $z^\circ$-covered ring then so is $S$. We prove that a surjective ring homomorphism $\phi:R\to S$ is strong if and only if ${\rm ker}(\phi)\subseteq{\rm rad}(R)$.

Let M be a module over a commutative ring R. We continue our study of strongly annihilating submodule graph SAG(M) introduced in [11]. In addition to providing the more properties of this graph, we introduce the subgraph SAG∗ (M) of SAG(M) and compare the properties of SAG∗ (M) with SAG(M) and AG(M) (the annihilating submodule graph of M introduced in [4]).

‎‎‎In this paper first we introduce right(resp‎. ‎left) induced hyperlattices and investigate some of their properties‎. ‎Especially‎ ‎a characterization of the smallest strongly regular relation for the class of distributive right/left induced hyperlattice is investigated‎. ‎Next we propose and study the generated hyperlattices from hyperlattices‎. ‎Finally‎, ‎the right induced hyperlattices of two Boolean lattices of four DNA bases and physico-chemical properties of amino acids of four DNA bases are investigated‎.

Let $R$ be a commutative ring with identity and let $M$ be an $R$-module‎. ‎The purpose of this paper is to introduce and investigate the submodules of an $R$-module $M$ which satisfy the dual of Property $\mathcal{A}$‎, ‎the dual of strong Property $\mathcal{A}$‎, ‎and the dual of proper strong Property $\mathcal{A}$‎. ‎Moreover‎, ‎a submodule $N$ of $M$ which satisfy Property $\mathcal{S_J(N)}$ and Property $\mathcal{I^M_J(N)}$ will be introduced and investigated‎.

Let $KG$ be the modular group algebra of a group $G$ over a field $K$ of characteristic $p>0$. Recently, we have seen the classification of group algebras $KG$ with upper Lie nilpotency index $t^{L}(KG)$ up to $10p-8$. In this paper, our aim is to classify the modular group algebra $KG$ with upper Lie nilpotency index $11p-9$, for $G'= \gamma_{2}(G)$ as an abelian group.

This paper studies $\lambda$-constacyclic codes and skew $\lambda$-constacyclic codes over the finite commutative non-chain ring $R=\mathbb{Z}_4+u\mathbb{Z}_4+u^2\mathbb{Z}_4$ with $u^3=0$ for $\lambda= (1+2u+2u^2)$ and $(3+2u+2u^2)$. We introduce distinct Gray maps and show that the Gray images of $\lambda$-constacyclic codes are cyclic, quasi-cyclic, and permutation equivalent to quasi-cyclic codes over $\mathbb{Z}_4$. It is also shown that the Gray images of skew $\lambda$-constacyclic codes are quasi-cyclic codes of length $2n$ and index 2 over $\mathbb{Z}_4$. Moreover, the structure of $\lambda$-constacyclic codes of odd length $n$ over the ring $R$ is determined and give some suitable examples.

Breakable semihypergroups, defined by a simple property: every non-empty subset of them is a subsemihypergroup. In this paper, we introduce a class of semihypergroups, in which every hyperproduct of $n$ elements is equal to a subset of the factors, called $\pi_n$-semihypergroups. Then, we prove that every semihypergroup of type $\pi_{2k}$, ($k\geq 2$) is breakable and every semihypergroup of type $\pi_{2k+1}$ is of type $\pi_3$. Furthermore, we obtain a decomposition of a semihypergroup of type $\pi_n$ into the cyclic group of order 2 and a breakable semihypergroup. Finally, we give a characterization of semi-symmetric semihypergroups of type $\pi_n$.

One of the important problem in finite groups theory is group characterization by specific property. Properties, such as element order, the set of element with the same order, etc. In this paper, we prove that Ree group $^2{}G_2(q)$, where $q\pm\sqrt{3q}+1$ is a prime number can be uniquely determined by its order and one conjugacy class size.