Journal of Algebra and Its Applications

In this paper, we generalize the results of [M. Sánchez-Mirafuentes and G. Villa–Salvador, Radical extensions for the Carlitz module, J. Algebra 398 (2014) 284–302] to rank one Drinfeld modules with class number one. We show that, in the present form, there does not exist a cogalois theory for Drinfeld modules of rank or class number larger than one. We also consider the torsion of the Carlitz module for the extension ${𝔽}_q(T)({\mathrm{\Lambda }}_{P^n})/{𝔽}_q(T)({\mathrm{\Lambda }}_P)$.

For Temperley–Lieb algebras of types $B$ and $D$, we construct their Gröbner-Shirshov bases and the corresponding standard monomials

In this paper, the ring of polynomials is studied in a systematic way through the theory of monoid rings. As a consequence, this study provides canonical approaches in order to find easy and rigorous proofs and methods for many facts on polynomials and formal power series; some of them as sample are treated in this paper. Besides the universal properties of the monoid rings and polynomial rings, a universal property for the formal power series rings is also established.

Let $({𝒜},\text{-}\otimes \text{-})$ be a symmetric monoidal closed exact category. This category is a natural framework to define the notions of purity and flatness. When ${𝒜}$ is endowed with an injective cogenerator with respect to the exact structure, we show that an object ${ℱ}$ in ${𝒜}$ is flat if and only if any conflation ending in ${ℱ}$ is pure. Furthermore, we prove a generalization of the Lambek Theorem (J. Lambek, A module is flat if and only if its character module is injective, Canad. Math. Bull. 7 (1964) 237–243) in ${𝒜}$. In the case ${𝒜}$ is a quasi-abelian category, we prove that ${𝒜}$ has enough pure injective objects.

Let $R$ be an arbitrary ring. We use a strict ${𝒲}{𝒳}$-resolution $X$ of a complex $K$ with finite $\widetilde{\#\mathrm{\text{-}}{𝒳}}$-projective dimension, where ${𝒳}$ denotes a subcategory of right $R$-modules closed under extensions and direct summands and admits an injective cogenerator ${𝒲}$, to define the $n$th relative cohomology functor ${\mathrm{\text{Ext}}}_{{𝒳}C}^n(K,-)$ as ${\mathrm{\text{H}}}^{-n}{\mathscr{H}}\mathrm{\text{om}}(X,-)$. If a complex $L$ has finite $\widetilde{\#\mathrm{\text{-}}{𝒴}}$-injective dimension, then one can use a dual argument to define a notion of a relative cohomology functor ${\mathrm{\text{Ext}}}_{{𝒞}Y}^n(-,L)$, where ${𝒴}$ is a subcategory of right $R$-modules closed under extensions and direct summands and admits a projective generator. Under several orthogonal conditions, we show that there exists an isomorphism ${\mathrm{\text{Ext}}}_{{{𝒳}C}^n}(K,L)\cong {\mathrm{\text{Ext}}}_{{𝒞}{𝒴}}^n(K,L)$ of relative cohomology groups for each $n\in \mathbb{Z}$. This result simultaneously encompasses a balance result of Holm on Gorenstein projective and injective modules, a balance result of Sather-Wagstaff, Sharif and White on Gorenstein projective and injective modules with respect to semidualizing modules, and a balance result of Liu on Gorenstein projective and injective complexes. In particular, as an application of this result, we extend the above balance result of Sather-Wagstaff, Sharif and White to the setting of complexes.

We present several naturally occurring classes of spectral spaces using commutative algebra on pointed monoids. For this purpose, our main tools are finite type closure operations and continuous valuations on monoids which we introduce in this work. In the process, we make a detailed study of different closure operations on monoids. We prove that the collection of continuous valuations on a topological monoid with topology determined by any finitely generated ideal is a spectral space.

Let $n\ge 2$ and $d\ge 0$ be arbitrary fixed integers. We prove that there exists a ring $R$ such that: (1) $R$ is a right $(n,d)$-ring; (2) $R$ is not a right $(n-1,t)$-ring for each non-negative integer $t$; (3) $R$ is not a right $(m,d-1)$-ring $($for $d\ge 1)$, for each non-negative integer $m$; (4) $R$ is a right $n$-coherent ring; (5) $R$ is not a right $(n-1)$-coherent ring. This shows the richness of right $(n,d)$-rings and right $n$-coherent rings, and, in particular, answers affirmatively a problem posed by Costa in [D. L. Costa, Parameterizing families of non-Noetherian rings, Comm. Algebra 22 (1994) 3997–4011.] when the ring in question is non-commutative.

Let $G$ be a group. Two elements $x,y\in G$ are said to be in the same $z$-class if their centralizers in $G$ are conjugate within $G$. In this paper, we prove that the number of $z$-classes in the group of upper triangular matrices is infinite provided that the field is infinite and size of the matrices is at least $6$, and finite otherwise.

The present work aims to exploit the interplay between the algebraic properties of rings and the graph-theoretic structures of their associated graphs. Let $R$ be an associative (not necessarily commutative) ring. We focus on the domination number of the zero-divisor graph $\mathrm{\Gamma }(R)$, the compressed zero-divisor graph ${\mathrm{\Gamma }}_E(R)$ and the unit graph $G(R)$. We find some relations between the domination number of the zero-divisor graph and that of the compressed zero-divisor graph. Moreover, some relations between the domination number of $\mathrm{\Gamma }(R)$ and $\mathrm{\Gamma }(R[x;\alpha ,\delta ])$, as well as the relations between the domination number of $G(R)$ and $G(R[[x;\alpha ]])$, are studied.

By applying an old result of Y. Berkovich, we provide a polynomial-time algorithm for computing the minimal possible index of a proper subgroup of a finite permutation group $G$. Moreover, we find that subgroup explicitly and within the same time if $G$ is given by a Cayley table. As a corollary, we get an algorithm for testing whether or not a finite permutation group acts on a tree non-trivially.

Let ${ℛ}$ be a ring. Motivated by a generalization of a well-known minus partial order to Rickart rings, we introduce a new relation on the power set ${𝒫}({ℛ})$ of ${ℛ}$ and show that this relation, which we call “the minus order on ${𝒫}({ℛ})$”, is a partial order when ${ℛ}$ is a Baer ring. We similarly introduce and study properties of the star, the left-star, and the right-star partial orders on the power sets of Baer ∗-rings. We show that some ideals generated by projections of a von Neumann regular and Baer ∗-ring ${ℛ}$ form a lattice with respect to the star partial order on ${𝒫}({ℛ})$. As a particular case, we present characterizations of these orders on the power set of $B({\mathscr{H}})$, the algebra of all bounded linear operators on a Hilbert space ${\mathscr{H}}$.

By considering the nilpotent Lie algebra with the derived subalgebra of dimension $2$, we compute some functors including the Schur multiplier, the exterior square and the tensor square of these Lie algebras. We also give the corank of such Lie algebras.

In this paper, we introduce the concept of $n$-$Lie$-isoclinism on non-Lie Leibniz algebras. Among the results obtained, we provide several characterizations of $n$-$Lie$-isoclinic classes of Leibniz algebras. Also, we provide a characterization of $n$-$Lie$-stem Leibniz algebras, and prove that every $n$-$Lie$-isoclinic class of Leibniz algebras contains a $n$-$Lie$-stem Leibniz algebra.

Let $f(x)=x^8+a{x}^4+b$ be an irreducible polynomial with rational coefficients, $K/Q$ the number field defined by $f$, and $G$ the Galois group of $f$. Let $g(x)=x^4+a{x}^2+b$, and let $G_4$ be the Galois group of $g$. We investigate the extent to which knowledge of the conjugacy class of $G_4$ in $S_4$ determines the conjugacy class of $G$ in $S_8$. We show that, in general, knowledge of $G_4$ does not automatically determine $G$, except when $G_4$ is isomorphic to $C_4$ (the cyclic group of order 4). In this case, we show $G$ is isomorphic to a non-split extension of $D_4$ (the dihedral group of order 8) by $C_4$. We also show that $G$ is completely determined when $G_4$ is isomorphic to $D_4$ and $4b-a^2$ is a perfect square. In this case, $G\simeq C_4\wr C_2\simeq (C_4\times C_4)⋊C_2$.

Given any minimal ring extension $k\subset L$ of finite fields, several families of examples are constructed of a finite local (commutative unital) ring $A$ which is not a field, with a (necessarily finite) inert (minimal ring) extension $A\subset B$ (so that $B$ is a separable $A$-algebra), such that $A\subset B$ is not a Galois extension and the residue field of $A$ (respectively, $B$) is $k$ (respectively, $L$). These results refute an assertion of G. Ganske and McDonald stating that if $R\subseteq S$ are finite local rings such that $S$ is a separable $R$-algebra, then $R\subseteq S$ is a Galois ring extension. We identify the homological error in the published proof of that assertion. Let $(A,M)$ be a finite special principal ideal ring (SPIR), but not a field, such that $M$ has index of nilpotency $\alpha$ ($\ge 2$). Impose the uniform distribution on the (finite) set of ($A$-algebra) isomorphism classes of the minimal ring extensions of $A$. If $2\in M$ (for instance, if $A\cong \mathbb{Z}/2^{\alpha }\mathbb{Z}$), the probability that a random isomorphism class consists of ramified extensions of $A$ is at least $2/3$; if $2\notin M$ (for instance, if $A\cong \mathbb{Z}/p^{\alpha }\mathbb{Z}$ for some odd prime $p$), the corresponding probability is at least $3/4$. Additional applications, examples and historical remarks are given.

The purpose of this paper is to study certain geometric properties of generalized quadrangle $(\mathrm{\Omega },𝔏)$ of type $O_6^{-}(2)$. We define certain root elements which generate a Lie algebra of type $E_6(K)$ for fields $K$ of characteristic two. The construction will be mainly based on the geometric properties of the generalized quadrangle $(\mathrm{\Omega },𝔏)$. In fact, we will explicity construct a Chevalley base of this Lie algebra.

We show in this note that if $B$ is a Boolean subalgebra of the lattice quantum logic $L$, then each state on $B$ can be extended over $L$ as a Jauch–Piron state provided $L$ is Jauch–Piron unital with respect to $B$ (i.e. for each nonzero $b\in B$, there is a Jauch–Piron state $s$ on $L$ such that $s(b)=1$). We then discuss this result for the case of $L$ being the Hilbert space logic $L(H)$ and $L$ being a set-representable logic.

Recall that a module $M$ is FI-extending if every fully invariant submodule is essential in a direct summand of $M$. Let $K=\left(\begin{array}{cc}\hfill R\hfill & \hfill M\hfill \\ \hfill N\hfill & \hfill S\hfill \end{array}\right)$ be a generalized matrix ring, where $R$ and $S$ are rings and ${}_R{M}_S$ and ${}_S{N}_R$ are bimodules. In this paper, we investigate necessary and/or sufficient conditions on $R$, $S$, $M$, and $N$ for $K_K$ to be FI-extending.

In this paper, we initiate the study of the notion of support of a hypermodule over a Krasner hyperring, providing several connections with the annihilator of such hypermodules. We concentrate on the similarities/differences between these concepts and the analogous ones from classical algebra (module theory). After defining and characterizing the support of a hypermodule (and in particular of a finitely generated hypermodule), by using the notion of length of a hypermodule, we determine the support of a quotient hypermodule containing only one maximal hyperideal.

The enhanced power graph of a finite group $G$ is the graph whose vertex set is $G$, and two distinct vertices are adjacent if they generate a cyclic subgroup of $G$. In this paper, we establish an explicit formula for the metric dimension of an enhanced power graph. As an application, we compute the metric dimension of the enhanced power graph of an elementary abelian $p$-group, a dihedral group and a generalized quaternion group.