Journal of Algebraic Systems
Volume:11 Issue: 2, Winter-Spring 2024

‎This work aims to introduce and investigate a preordering in $B(\mathcal{H}),$‎ ‎the Banach space of all bounded linear operators defined on a complex‎ ‎Hilbert space $\mathcal{H}.$ It is called strong majorization and denoted by $S\prec_{s}T,$ for‎ ‎$S,T\in B(\mathcal{H}).$ The strong majorization follows majorization defined by Barnes‎, ‎but not vice versa‎. ‎If $S\prec_{s}T,$ then $S$ inherits some properties of $T.$ ‎‎‎‎ The strong majorization will be extended for the d-tuple of operators in $B(\mathcal{H})^{d}$ and‎ ‎is called joint strong majorization denoted by $S\prec_{js}T,$ for $S,T\in B(\mathcal{H})^{d}.$ We show that‎ ‎some properties of strong majorization are satisfied for joint strong majorization‎.

In this paper, we introduce and study isotonic closure functions on a locale. These are pairs of the form $(L, \underline{{\mathrm{cl}}}_L)$, where$L$ is a locale and $\underline{{\mathrm{cl}}}_L\colon \mathcal{S}\!\ell(L) \rightarrow \mathcal{S}\!\ell(L)$is an isotonic closure function on the sublocales of $L$. Moreover, we introduce generalized$\underline{{\mathrm{cl}}}_L$- closed sublocales in isotonic closure locales and discuss some of their properties. Also, we introduce and study the category $ \textbf{ICF} $ whose objects and morphisms are isotonic closure functions $(L, \underline{{\mathrm{cl}}}_L)$ and localic maps, respectively.

‎T‎‎‎‎he main purpose of this article is to ‎present a‎ ‎new ‎approach ‎to ‎the‎ classification of all indecomposable absorbing ‎prime‎ multiplication modules with finite-dimensional top over pullback rings of two Dedekind ‎domains. First‎, ‎we give a complete description of the absorbing ‎prime ‎multiplication modules over a local Dedekind ‎domain‎. ‎‎‎In fact‎, ‎we extend the definition and results given in \cite{108} to a more general absorbing ‎prime‎ multiplication modules ‎case‎‎. ‎Next‎, ‎we‎ establish a connection between the absorbing ‎prime ‎multiplication modules and the pure-injective modules over such‎ ‎rings‎.

Let $R$ be an associative ring and $M$ be a monoid‎. ‎In this paper‎, ‎we introduce new kind of graph structure asociated with zero-divisors of monoid ring $R[M]$‎, ‎calling it the $M$-Armendariz graph of a ring $R$ and denoted by $A(R,M)$‎. ‎It is an undirected graph whose vertices are all non-zero zero-divisors of the monoid ring $R[M]$ and two distinct vertices $\alpha=a_{1}g_{1}+\cdots‎+ ‎a_{n}g_{n}$ and $\beta=b_{1}h_{1}+\cdots+b_{m}h_{m}$ are adjacent if and only if $a_{i}b_{j}=0$ or $b_{j}a_{i}=0$ for all $i,j$‎. ‎We investigate some graph properties of $A(R,M)$ such as diameter‎, ‎girth‎, ‎domination number and planarity‎. ‎Also‎, ‎we get some relations between diameters of the $M$-Armendariz graph $A(R,M)$ and that of zero divisor graph $\Gamma(R[M])$‎, ‎where $R$ is a reversible ring and $M$ is a unique product monoid‎.

The notion of fuzzy ideal in bounded equality algebras is defined, and several properties are studied. Fuzzy ideal generated by a fuzzy set is established, and a fuzzy ideal is made by using the collection of ideals. Characterizations of fuzzy ideal are discussed. Conditions for a fuzzy ideal to attains its infimum on all ideals are provided. Homomorphic image and preimage of fuzzy ideal are considered. Finally, quotient structures of equality algebra induced by (fuzzy) ideal are studied.

It is known that for a discrete valuation v of a field K with value group Z, an valued extension field (K′, v′) of (K, v) is generated by a root of an Eisenstein polynomial with respect to v having coefficients in K if and only if the extension (K′, v′)/(K, v) is totally ramified. The aim of this paper is to present the analogue of this result for valued field extensions generated by a root of an Eisenstein-Dumas polynomial with respect to a more general valuation (which is not necessarily discrete). This leads to classify such algebraic extensions of valued fields.

Let R be a commutative ring with identity 1≠0. The comaximal ideal graph of R is the simple, undirected graph whose vertex set is the set of all proper ideals of the ring R not contained in Jacobson radical of R and two vertices I and J are adjacent in this graph if and only if I+J=R. In this article, we have discussed the graph G(R) whose vertex set is the set of all proper ideals of ring R and two vertices I and J are adjacent in this graph if and only if I+J≠R. In this article, we have discussed some interesting results about G(R) and its line graph.

In this paper we generalize the Rigidity Theorem and Zero Divisor Conjecture for an arbitrary Serre subcategory of modules. For this purpose, for any regularM-sequence x1; :::; xn with respect to S we prove that if TorR 2 ((x1;:::;x R n); M) 2 S, thenTorR i ((x1;:::;x R n); M) 2 S, for all i ≥ 1. Also we show that if Extn R+2((x1;:::;x R n); M) 2 S,then Exti R((x1;:::;x R n); M) 2 S, for all integers i ≥ 0 (i ̸= n).

In this paper, we introduce the concept of uniformly $n$-ideal ofcommutative rings which is a special type of $n$-ideal. We call aproper ideal $I$ of $R$ a uniformly $n$-ideal if there exists apositive integer $k$ for $a,b\in R$ whenever $ab\in I$ and$a\notin I$ implies that $b^{k}=0.$ The basic properties ofuniformly $n$-ideals are investigated in detail. Moreover, somecharacterizations of uniformly $n$-ideals are obtained for somespecial rings.

Let R be a commutative Noetherian ring, I, J be ideals of R such thatJ ⊆ I, and M be a finitely generated R-module. In this paper, we prove that theinvariants AJI(M) := inf{i ∈ N0 | JtHiI (M) is not Artinian for all t ∈ N0} and inf{i ∈N0 | JtHiI (M) is not minimax for all t ∈ N0} are equal. In particular, we show that theinvariants AII(M) and inf{i ∈ N0 | HiI (M) is not minimax} are equal. We also establishthe local-global principle, AJI(M) = inf{AJRpIRp(Mp)|p ∈ Spec (R)}, in some cases.

Let $S=K[x_1,\ldots,x_n]$ be a polynomial ring over a field $K$ and$I\subset S$ be a monomial ideal with a linearresolution. Let$\frak{m}=(x_1,\ldots,x_n)$ be the unique homogeneous maximal ideal and $I\frak{m}$ be apolymatroidal ideal. We prove that if either $I\frak{m}$ is polymatroidal with strongexchange property, or $I$ is a monomial ideal in at most 4variables, then $I$ is polymatroidal. We also show that the firsthomological shift ideal of polymatroidal ideal is againpolymatroidal.

The rings considered in this article are commutative with identity which are not integral domains. Let R be a ring. The sum annihilating ideal graph of R is an undirected graph whose vertex set is the set of all non-zero annihilating ideals of R and distinct vertices I and J are adjacent if and only if their sum is an annihilating ideal. The aim of this article is to discuss some results on the domination number of the sum annihilating ideal graph of R and on the domination number of its complement.