Journal of Algebra and Its Applications

Let $R$ be a ring and ${𝒬}$ be a finite quiver. We give an explicit formula for the injective envelopes and projective precovers in the category $\mathrm{\text{Rep}}({{𝒬}}_I,R)$ of bound representations of ${𝒬}$ by left $R$-modules, where $I$ is a combination of monomial and commutativity relations. Some applications will be provided. In particular, it is shown that if ${𝒬}$ is acyclic and $R$ is an Iwanaga-Gorenstein ring, then so are these bound quiver algebras.

We present a linear algebra algorithm, which, given a zero-dimensional ideal via a “good” representation, produces a separating linear form, its radical, again given via the same “good” representation and also a Kronecker parametrization of it. As an irrelevant byproduct, if you like, you can also read a Gröbner basis of the radical.

An ideal $I$ of a ring $R$ is called pseudo-irreducible if $I$ cannot be written as an intersection of two comaximal proper ideals of $R$. In this paper, it is shown that the maximal spectrum of $R$ is Noetherian if and only if every proper ideal of $R$ can be expressed as a finite intersection of pseudo-irreducible ideals. Using a result of Hochster, we characterize all $T_1$ quasi-compact Noetherian topological spaces.

This work deals with the notion of Newton complementary duality as raised originally in the work of the second author and B. Costa. A conceptual revision of the main steps of the notion is accomplished which then leads to a vast simplification and improvement of several statements concerning rational maps and their images. A ring-homomorphism like map is introduced that allows for a close comparison between the respective graphs of a rational map and its Newton dual counterpart.

Necessary and sufficient conditions for a group to possess a faithful irreducible representation are investigated. We consider locally finite groups and groups whose socle is essential.

Let $S$ be a Suzuki group ${}^2{B}_2(q^2)$, where $q^2=2^{2f+1}$, $f\ge 1$. In this paper, we determine the degrees of the ordinary complex irreducible characters of every group $G$ such that $S\le G\le \mathrm{\text{Aut}}(S)$.

We introduce a certified numerical algorithm to calculate Lagrange’s resolvent. This algebraic certification uses Galoisian ideal.

Our goal is to determine when the trivial extensions of commutative rings by modules are Cohen–Macaulay in the sense of Hamilton and Marley. For this purpose, we provide a generalization of the concept of Cohen–Macaulayness of rings to modules.

We completely determine the rings for which every element is a sum of a nilpotent, an idempotent and a tripotent that commute with one another, and the rings for which every element is a sum of a nilpotent and two tripotents that commute with one another.

In this work, we compute solutions of the Yang–Baxter associative equation in dimensions one and two. For these solutions, we describe the double constructions of the associated Frobenius algebras, following Bai’s definitions. Besides, we determine related compatible dendriform algebras and solutions of their $D-$equations. Finally, using symmetric solutions of these equations, we build the double constructions of related Connes cocycles.

We use purely combinatorial arguments to give a formula to compute all graded Betti numbers of path ideals of paths and cycles. As a consequence, we can give new and short proofs for the known formulas of regularity and projective dimensions of path ideals of paths.

A morphism $f:M\to N$ of left $R$-modules is called an $n$-phantom morphism if the induced morphism ${\mathrm{\text{Tor}}}_n^R(A,f)=0$ for every (finitely presented) right $R$-module $A$. Analogously, a morphism $g:X\to Y$ of left $R$-modules is said to be an $n$-$\mathrm{\text{Ext}}$-phantom morphism if the induced morphism ${\mathrm{\text{Ext}}}_R^n(B,g)=0$ for every finitely presented left $R$-module $B$. We study the properties of $n$-phantom and $n$-Ext-phantom morphisms. Some applications are given.

The usual commutative ideal theory was extended to ideals in noncommutative rings by Lambek, introducing the concept of symmetric. Camillo et al. naturally extended the study of symmetric ring property to the lattice of ideals, defining the new concept of an ideal-symmetric ring. This paper focuses on the symmetric ring property on nil ideals, as a generalization of an ideal-symmetric ring. A ring $R$ will be said to be right (respectively, left) nil-ideal-symmetric if $IJK=0$ implies $IKJ=0$ (respectively, $JIK=0$) for nil ideals $I,J,K$ of $R$. This concept generalizes both ideal-symmetric rings and weak nil-symmetric rings in which the symmetric ring property has been observed in some restricted situations. The structure of nil-ideal-symmetric rings is studied in relation to the near concepts and ring extensions which have roles in ring theory.

In this paper, we introduce and study left (right) $G$-semihereditary rings over any associative ring, and these rings are exactly $G$-semihereditary rings defined by Mahdou and Tamekkante provided that $R$ is a commutative ring. Some new characterizations of left $G$-semihereditary rings are given. Applications go in three directions. The first is to give a sufficient condition when a finitely presented right $R$-module is Gorenstein flat if and only if it is Gorenstein projective provided that $R$ is left coherent. The second is to investigate the relationships between Gorenstein flat modules and direct limits of finitely presented Gorenstein projective modules. The third is to obtain some new characterizations of semihereditary rings, $1$-$FC$ rings and $IF$ rings.

A reality-based algebra (RBA) is a finite-dimensional associative algebra with involution over $ℂ$ whose distinguished basis $B$ contains $1$ and is closed under pseudo-inverse. An integral RBA is one whose structure constants in its distinguished basis are integers. If the algebra has a one-dimensional representation taking positive values on $B$, then we say that the RBA has a positive degree map. These RBAs have a standard feasible trace, and the multiplicities of the irreducible characters in the standard feasible trace are the multiplicities of the RBA. In this paper, we show that for integral RBAs with positive degree map whose multiplicities are rational, any finite subgroup of torsion units whose elements are all of degree $1$ and have algebraic integer coefficients must have order dividing a certain positive integer determined by the degree map and the multiplicities. The paper concludes with a thorough investigation of the properties of RBAs that force multiplicities to be rational.

In this work, we consider the existence and construction of pseudocomplements in some lattices of module classes. The classes of modules belonging to these lattices are defined via closure under operations such as taking submodules, quotients, extensions, injective hulls, direct sums or products. We characterize the rings for which the lattices $R$-tors (of hereditary torsion classes), $R$-nat (the lattice of natural classes) and $R$-conat (the lattice of conatural classes) coincide.

This work introduces a new kind of semigroup of ${\mathbb{N}}^p$ called proportionally modular affine semigroup. These semigroups are defined by modular Diophantine inequalities and they are a generalization of proportionally modular numerical semigroups. We give an algorithm to compute their minimal generating sets, and we specialize when $p=2$. For this case, we also provide a faster algorithm to compute their minimal system of generators, prove they are Cohen–Macaulay and Buchsbaum, and determinate their (minimal) Frobenius vectors. Besides, Gorenstein proportionally modular affine semigroups are characterized.

When ${𝒞}$ is a chordal clutter in the sense of Woodroofe or Emtander, we show that the complement clutter is edgewise strongly shellable. When ${𝒞}$ is indeed a finite simple graph, we provide additional characterization of chordal graphs from the point of view of strong shellability. In particular, the generic graph $G_T$ of a tree is shown to be bi-strongly shellable.