Algebra Colloquium

In this paper, we describe the semigroups satisfying the implication e²=e⇒exyey=exy and answer an open question raised by Guo, Shum and Zhu in 1999.

In this article, we give a short proof for the existence and uniqueness of the Higman–Sims sporadic simple group 𝖧𝖲 by means of the first author's algorithm [17] and uniqueness criterion [18], respectively. We realize 𝖧𝖲 as a subgroup of GL₂₂(11), and determine its automorphism group Aut(𝖧𝖲). We also give a presentation for Aut(𝖧𝖲) in terms of generators and relations. Furthermore, the character table of 𝖧𝖲 is determined and representatives of its conjugacy classes are given as short words in its generating matrices inside GL₂₂(11).

For an endomorphism σ of a ring R, Krempa called σ a rigid endomorphism if aσ(a)=0 implies a=0 for a∈R. A ring R is called rigid if there exists a rigid endomorphism of R. In this paper, we extend the σ-rigid property of a ring R to an ideal of R. For a σ-ideal I of a ring R, we call I a σ-rigid ideal if aσ(a)∈I implies a∈I for a∈R. We characterize σ-rigid ideals and study related properties. The connections of the prime radical and the upper nil radical of R with the prime radical and the upper nil radical of the Ore extension R[x;σ,δ], respectively, are also investigated.

In this paper, we introduce the little q-Schur algebra s(2,r) of type A₁. The main purpose of this paper is to construct a monomial base of s(2,r).

If G is a finite group, we define its prime graph Γ(G) as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, q are joined by an edge, denoted by p~q, if there is an element in G of order pq. Assume with primes p₁<p₂<⋯<p_k and natural numbers α_i. For p∈π(G), let the degree of p be deg(p)=|{q∈π(G)|q~p}|, and D(G):=(deg(p₁), deg(p₂),…, deg(p_k)). In this paper, we prove that if G is a finite group such that D(G)=D(M) and |G|=|M|, where M is one of the following simple groups: (1) sporadic simple groups, (2) alternating groups A_p with p and p-2 primes, (3) some simple groups of Lie type, then G≅M. Moreover, we show that if G is a finite group with OC(G)={2⁹.3⁹.5.7, 13}, then G≅S₆(3) or O₇(3), and finally, we show that if G is a finite group such that |G|=2⁹.3⁹.5.7.13 and D(G)=(3,2,2,1,0), then G≅S₆(3) or O₇(3).

The purpose of this paper is to establish connection between the minimal flat resolutions of Artinian modules and the concept of cosequences in commutative algebra. The main result of this paper leads to a characterization of local Cohen–Macaulay rings in terms of vanishing property of the dual Bass numbers of certain Artinian modules.

In this paper, (generalized) soluble groups for which the set of non-modular subgroups verifies the maximal condition and groups for which the set of non-permutable subgroups satisfies the same property are classified.

Let D be a division ring with centre F. Assume that M is a maximal subgroup of GL_n(D) (n≥1) such that Z(M) is algebraic over F. Group identities on M and polynomial identities on the F-linear hull F[M] are investigated. It is shown that if F[M] is a PI-algebra, then [D:F]<∞. When D is non-commutative and F is infinite, it is also proved that if M satisfies a group identity and F[M] is algebraic over F, then we have either M=K* where K is a field and [D:F]<∞, or M is absolutely irreducible. For a finite dimensional division algebra D, assume that N is a subnormal subgroup of GL_n(D) and M is a maximal subgroup of N. If M satisfies a group identity, it is shown that M is abelian-by-finite.

In crisp environment, the notion of cyclic group on a set is well known. We study an extension of this classical notion to the fuzzy sets to define the concept of cyclic fuzzy subgroups. By using these cyclic fuzzy subgroups, we then define a cyclic fuzzy group family and investigate its structure properties.

Let (S,≤) be a strictly ordered monoid, and R a right noetherian ring. Assume that M is a finitely generated right R-module and N a left R-module. Denote by [[M^S,≤]] (resp., [[N^S,≤]]) the right (resp., left) [[R^S,≤]]-module of generalized power series over M (resp., over N). Then we show that there exists an isomorphism of abelian groups .

In this paper, we introduce the notion of Δ′-reflexive modules. We investigate the reflexive modules with respect to a partial cotilting (bi)module and obtain a connection between Δ′-reflexive modules and Cogen(P)-linearly compact modules. The main results generalize the results on reflexive modules with respect to a cotilting bimodule.

A corrected and completed list of six dimensional real Lie algebras with five dimensional nilradical is presented. Their invariants for the coadjoint representation are computed and some results on the invariants of solvable Lie algebras in arbitrary dimension whose nilradical has codimension one are also given. Specifically, it is shown that any rank one solvable Lie algebra of dimension n without invariants determines a family of (n+2k)-dimensional algebras with the same property.

A hypersubstitution is a map which takes n-ary operation symbols to n-ary terms. Any such map can be uniquely extended to a map defined on the set W_τ(X) of all terms of type τ, and any two such extensions can be composed in a natural way. Thus, the set Hyp(τ) of all hypersubstitutions of type τ forms a monoid. In this paper, we characterize Green's relation ℛ on the monoid Hyp(τ) for the type τ=(n,n). In this case, the monoid of all hypersubstitutions is isomorphic with the monoid of all clone endomorphisms. The results can be applied to mutually derived varieties.

Let bcl(G) denote the largest conjugacy class length of a finite group G. In this note, we prove that if bcl(G)<p² for a prime p, then |G:O_p(G)|_p≤p.

In this paper, we investigate the structure of loops in which the associativity relation between pairs of non-trivial elements (i.e., 〈x,y〉 is a group) is transitive. We also look at other properties that come out of the study of this class of loops.