Journal of Algebra and Its Applications

In this paper, we introduce and investigate the latticial counterparts of the conditions (C_i), i = 1, 2, 3, 11, 12, for modules. In particular, we study the lattices satisfying the condition (C₁), we call CC lattices (for Closed are Complements), i.e. the lattices such that any closed element is a complement, that are the latticial counterparts of CS modules (for Closed are Summands). Applications of these results are given to Grothendieck categories and module categories equipped with a torsion theory.

We say a ring R with unity is weakly principally quasi-Baer or simply (weakly p.q.-Baer) if the right annihilator of a principal right ideal is right s-unital by right semicentral idempotents, which implies that R modulo the right annihilator of any principal right ideal is flat. It is proven that the weakly p.q.-Baer property is inherited by polynomial extensions and includes both left p.q.-Baer rings and right p.q.-Baer rings and is closed under direct products and Morita invariance. A ring R is weakly p.q.-Baer if and only if the upper triangular matrix ring T_n(R) is weakly p.q.-Baer. We extend a theorem of Kist for commutative Baer rings to weakly p.q.-Baer rings for which every prime ideal contains a unique minimal prime ideal without using topological arguments. It is also shown that there is an important subclass of weakly p.q.-Baer rings which have a nontrivial subdirect product representation.

Let m^NEF(n) be the smallest positive integer p such that the dimensions of global sections of multiple adjoint bundles h⁰(m(K_X + L)) are strictly greater than zero for any integer m with m ≥ p and any quasi-polarized n-folds (X, L) for which X is a complex normal Gorenstein projective variety with only rational singularities and K_X + L is nef. In this paper, we prove that m^NEF(n) ≤ 2n - 4 holds for n ≥ 4.

For a simplicial complex Δ, we introduce a simplicial complex attached to Δ, called the expansion of Δ, which is a natural generalization of the notion of expansion in graph theory. We are interested in knowing how the properties of a simplicial complex and its Stanley–Reisner ring relate to those of its expansions. It is shown that taking expansion preserves vertex decomposable and shellable properties and in some cases Cohen–Macaulayness. Also it is proved that some homological invariants of Stanley–Reisner ring of a simplicial complex relate to those invariants in the Stanley–Reisner ring of its expansions.

We indicate a strategy in order to construct bilinear multiplication algorithms of type Chudnovsky in large extensions of any finite field. In particular, using the symmetric version of the generalization of Randriambololona specialized on the elliptic curves, we show that it is possible to construct such algorithms with low bilinear complexity. More precisely, if we only consider the Chudnovsky-type algorithms of type symmetric elliptic, we show that the symmetric bilinear complexity of these algorithms is in where n corresponds to the extension degree, and is the iterated logarithm. Moreover, we show that the construction of such algorithms can be done in time polynomial in n. Finally, applying this method we present the effective construction, step by step, of such an algorithm of multiplication in the finite field 𝔽_3⁵⁷.

As the 3-string braid group B₃ and the modular group Γ are both of wild representation type one cannot expect a full classification of all their finite dimensional simple representations. Still, one can aim to describe 'most' irreducible representations by constructing for each d-dimensional irreducible component X of the variety iss_n(Γ) classifying the isomorphism classes of semi-simple n-dimensional representations of Γ an explicit minimal étale rational map 𝔸^d → X having a Zariski dense image. Such rational dense parametrizations were obtained for all components when n < 12 in [5]. The aim of the present paper is to establish such parametrizations for all finite dimensions n.

In this paper, we find the ω-value of the generators of any numerical semigroup with embedding dimension three. This allows us to determine all possible orderings of the ω-values of the generators. In addition, we relate the ω-value of the numerical semigroup to its catenary degree.

The main aim of this paper is to investigate two characteristic subgroups ω^𝒜(G) and θ^𝒜(G) of a finite group G, which are defined as the intersections of the normalizers of derived subgroups of subnormal and non-subnormal subgroups of G respectively. Our main theory improve and extend some earlier results.

The content of a polynomial over a ring R is a well-understood notion. Ohm and Rush generalized this concept of a content map to an arbitrary ring extension of R, although it can behave quite badly. We examine five properties an algebra may have with respect to this function — content algebra, weak content algebra, semicontent algebra (our own definition), Gaussian algebra, and Ohm–Rush algebra. We show that the Gaussian, weak content, and semicontent algebra properties are all transitive. However, transitivity is unknown for the content algebra property. We then compare the Ohm–Rush notion with the more usual notion of content in the power series context. We show that many of the given properties coincide for the power series extension map over a valuation ring of finite dimension, and that they are equivalent to the value group being order-isomorphic to the integers or the reals. Along the way, we give a new characterization of Prüfer domains.

Let A₁,…,A_n(n ≥ 2) be ideals of a commutative ring R. Let G(k) (resp., L(k)) denote the product of all the sums (resp., intersections) of k of the ideals. Then we have

In this paper we prove a sufficient condition for the existence of matchings in arbitrary groups and its linear analogue, which lead to some generalizations of the existing results in the theory of matchings in groups and central extensions of division rings. We introduce the notion of relative matchings between arrays of elements in groups and use this notion to study the behavior of matchable sets under group homomorphisms. We also present infinite families of prime numbers p such that ℤ/pℤ does not have the acyclic matching property. Finally, we introduce the linear version of acyclic matching property and show that purely transcendental field extensions satisfy this property.

We introduce and study zero-divisor graphs in categories of left modules over a ring R, i.e. R-MOD. The vertices of Γ(R-MOD) consist of all nonzero morphisms in R-MOD which are not isomorphisms. Two vertices f and g are adjacent if f ◦ g = 0 or g ◦ f = 0. We observe that these graphs are connected and their diameter is equal or less than four. We prove that Γ(R-MOD) = 3 if and only if R is a right and left perfect ring and R/J(R) is simple artinian. We also characterize all vertices with complements and that when a kernel or a co-kernel can be a complement for a morphism. Some discussions will be made on radius of these graphs, their clique and chromatic numbers.

We investigate the Zassenhaus Conjecture for the integral group ring of the simple group PSL(3, 4).

Let R be a commutative ring with identity and let G(V, E) be a graph. The locating number of the graph G(V, E) denoted by loc(G) is the cardinality of the minimal locating set W ⊆ V(G). To get the loc(G), we assign locating codes to the vertices V(G)∖W of G in such a way that every two vertices get different codes. In this paper, we consider the ratio of loc(G) to |V(G)| and show that there is a finite connected graph G with loc(G)/|V(G)| = m/n, where m < n are positive integers. We examine two equivalence relations on the vertices of Γ(R) and the relationship between locating sets and the cut vertices of Γ(R). Further, we obtain bounds for the locating number in zero-divisor graphs of a commutative ring and discuss the relation between locating number, domination number, clique number and chromatic number of Γ(R). We also investigate the locating number in Γ(R) when R is a finite product of rings.

Let G be a finite group. A subgroup A of G is called a TI-subgroup of G if A ∩ A^x = 1 or A for all x ∈ G. A subgroup H of G is called a QTI-subgroup if C_G(x) ⊆ N_G(H) for every 1 ≠ x ∈ H, and a group G is called an MCTI-group if all its metacyclic subgroups are QTI-subgroups. In this paper, we show that every nilpotent MCTI-group is either a Dedekind group or a p-group and we completely classify all the MCTI-p-groups. We show that all MCTI-groups are solvable and that every nonnilpotent MCTI-group must be a Frobenius group having abelian kernel and cyclic complement.

In this paper, we characterize maximal non-ACCP subrings R of a domain S in case (R, S) is a residually algebraic pair and R is semilocal. In this paper, we also consider a K-algebra S, a nonzero proper ideal I of S and a subring D of the field K and we determine necessary and sufficient conditions in order that D + I is a maximal non-ACCP subring of S. This gives an example of a maximal non-ACCP subring R of a domain S such that (R, S) is a normal pair and R is not semilocal.

Let S = K[x₁,…,x_m, y₁,…,y_n] be the standard bigraded polynomial ring over a field K, and M a finitely generated bigraded S-module. In this paper we study the generalized Cohen–Macaulayness and sequentially generalized Cohen–Macaulayness of M with respect to Q = (y₁,…,y_n). We prove that if I ⊆ S be a monomial ideal with cd(Q, S/I) ≤ 2, then S/I is sequentially generalized Cohen–Macaulay with respect to Q.

It is known that the ranks of the semigroups 𝒮ing_n and 𝒮𝒫𝒯_n (the semigroups of singular self-maps and partial transformations on X_n = {1, 2, …, n}, respectively) are , respectively. The idempotent ranks, defined as the smallest number of idempotents generating set, of 𝒮ing_n and 𝒮𝒫𝒯_n are the same value as the ranks, respectively. Idempotent can be seen as a special case (with m = r = 1) of (m, r)-potent. In this paper, we determine the (m, r)-potent ranks, defined as the smallest number of (m, r)-potent generating set, of the semigroups 𝒮ing_n and 𝒮𝒫𝒯_n.

Let R be a commutative Noetherian ring, 𝔞 an ideal of R and M, N two finitely generated R-modules. Let t be a positive integer or ∞. We denote by Ω_t the set of ideals 𝔠 such that for all i < t. First, we show that there exists the ideal 𝔟_t which is the largest in Ω_t and . Next, we prove that if 𝔡 is an ideal such that 𝔞 ⊆ 𝔡 ⊆ 𝔟_t, then for all i < t.