Algebra Colloquium

We prove that a finite group G is p-supersolvable or p-nilpotent if some subgroups of G are weakly s-semipermutable in G. Several earlier results are generalized.

The associated primes of an arbitrary lexsegment ideal I ⊆ S=K[x₁,…,x_n] are determined. As application it is shown that S/I is a pretty clean module, therefore S/I is sequentially Cohen-Macaulay and satisfies Stanley's conjecture.

In this paper, we study an infinite-dimensional Lie algebra ℬ_q, called the q-analog Klein bottle Lie algebra. We show that ℬ_q is a finitely generated simple Lie algebra with a unique (up to scalars) symmetric invariant bilinear form. The derivation algebra and the universal central extension of ℬ_q are also determined.

We compute the minimal primary decomposition for completely squarefree lexsegment ideals. We show that critical squarefree monomial ideals are sequentially Cohen-Macaulay. As an application, we give a complete characterization of the completely squarefree lexsegment ideals which are sequentially Cohen-Macaulay and we also derive formulas for some homological invariants of this class of ideals.

A finite Gröbner-Shirshov basis is constructed for the plactic algebra of rank 3 over a field K. It is also shown that plactic algebras of rank exceeding 3 do not have finite Gröbner-Shirshov bases associated to the natural degree-lexicographic ordering on the corresponding free algebra. The latter is in contrast with the case of a strongly related class of algebras, called Chinese algebras.

Let I, J be ideals of a commutative Noetherian local ring (R, 𝔪) and let M be a finite R-module. The f-depth of M with respect to I is the least integer r such that is not Artinian. In this paper we show that is the least integer such that the local cohomology module with respect to a pair of ideals I, J is not Artinian. As a consequence, it follows that is (I,J)-cofinite for all . In addition, we show that for a Serre subcategory 𝖲, if belongs to 𝖲 for all i > n and if 𝔟 is an ideal of R such that belongs to 𝖲, then the module belongs to 𝖲.

Let M be a non-zero finitely generated module over a commutative Noetherian local ring (R, 𝔪). In this paper we consider when the local cohomology modules are finitely generated. It is shown that if t ≥ 0 is an integer and , then is not 𝔭-cofinite. Then we obtain a partial answer to a question raised by Huneke. Namely, if R is a complete local ring, then is finitely generated if and only if 0 ≤ n ∉ W, where . Also, we show that if J ⊆ I are 1-dimensional ideals of R, then is J-cominimax, and is finitely generated (resp., minimax) if and only if is finitely generated for all (resp., ). Moreover, the concept of the J-cofiniteness dimension of M relative to I is introduced, and we explore an interrelation between and the filter depth of M in I. Finally, we show that if R is complete and dim M/IM ≠ 0, then .

The concept of an I-matrix in the full 2 × 2 matrix ring M₂(R/I), where R is an arbitrary UFD and I is a nonzero ideal in R, is introduced. We obtain a concrete description of the centralizer of an I-matrix in M₂(R/I) as the sum of two subrings 𝒮₁ and 𝒮₂ of M₂(R/I), where 𝒮₁ is the image (under the natural epimorphism from M₂(R) to M₂(R/I)) of the centralizer in M₂(R) of a pre-image of , and the entries in 𝒮₂ are intersections of certain annihilators of elements arising from the entries of . It turns out that if R is a PID, then every matrix in M₂(R/I) is an I-matrix. However, this is not the case if R is a UFD in general. Moreover, for every factor ring R/I with zero divisors and every n ≥ 3, there is a matrix for which the mentioned concrete description is not valid.

A graph 𝛤 is called G-locally-quasiprimitive if each normal subgroup of G_v acts on 𝛤(v) trivially or transitively for every vertex v. In this paper we analyse the global action and the structural information of such groups G, extending the previous results for locally-primitive graphs and vertex-transitive locally-quasiprimitive graphs.

In this paper we study some properties of the Hurwitz series ring HA over a commutative ring A, such as the nilradical of HA and the chain condition on its annihilators. We provide an example showing that the last property does not pass from A to HA. A strongly Hopfian ring is a ring satisfying the chain condition on some type of annihilators. We give a large class of strongly Hopfian rings A such that HA are not strongly Hopfian.

We study “Fibonacci type” groups and semigroups. By establishing asphericity of their presentations we show that many of the groups are infinite. We combine this with Adjan graph techniques and the classification of the finite Fibonacci semigroups (in terms of the finite Fibonacci groups) to extend it to the Fibonacci type semigroups.

For n ∈ ℕ, let O_n be the semigroup of all singular order-preserving mappings on [n]= {1,2,…,n}. For each nonempty subset A of [n], let O_n(A) = {α ∈ O_n: (∀ k ∈A) kα ≤ k} be the semigroup of all order-preserving and A-decreasing mappings on [n]. In this paper it is shown that O_n(A) is an abundant semigroup with n-1𝒟*-classes. Moreover, O_n(A) is idempotent-generated and its idempotent rank is 2n-2- |A\ {n}|. Further, it is shown that the rank of O_n(A) is equal to n-1 if 1 ∈ A, and it is equal to n otherwise.

Let (R, 𝔪) be a Noetherian local ring. Denote by N-dim_RA the Noetherian dimension of an Artinian R-module A. In this paper, we give some characterizations for the ring R to satisfy N-dim_RA =dim (R/Ann_RA) for certain Artinian R-modules A. Then the existence of a co-localization compatible with Artinian R-modules is studied and it is shown that if it is compatible with local cohomologies of finitely generated modules, then the base ring is universally catenary and all of its formal fibers are Cohen-Macaulay.

Let R be a commutative ring and M be an R-module. The second spectrum Spec^s(M) of M is the collection of all second submodules of M. We topologize Spec^s(M) with Zariski topology, which is analogous to that for Spec(M), and investigate this topological space. For various types of modules M, we obtain conditions under which Spec^s(M) is a spectral space. We also investigate Spec^s(M) with quasi-Zariski topology.

In this paper we give a characterization for certain HNN extensions of subgroup separable groups with normal associated subgroups to be weakly potent. We then apply our result to show that certain HNN extensions of finitely generated nilpotent groups with central associated subgroups are weakly potent.

In this paper, we extend the concept of Ako and Oka families to submodules, study the behavior of the extended prime submodule principle and use these concepts to give new proofs of some familiar theorems.

Let K be a normal subgroup of the finite group H. To a point of a K-interior H-algebra we associate a group extension, and we prove that this extension is isomorphic to an extension associated to the point given by the Brauer homomorphism. This may be regarded as a generalization and an alternative treatment of Dade's results [2, Section 12].