Algebra Colloquium

For a Specht module S^λ for the symmetric group Σ_d, the cohomology Hⁱ(Σ_d,S^λ) is known only in degree i = 0. We give a combinatorial criterion equivalent to the nonvanishing of the degree i = 1 cohomology, valid in odd characteristic. Our condition generalizes James' solution in degree zero. We apply this combinatorial description to give some computations of Specht module cohomology, together with an explicit description of the corresponding modules. Finally, we suggest some general conjectures that might be particularly amenable to proof using this description.

In this paper we introduce the notion of width in dimension > s for Artinian modules and give a finiteness result for attached primes of certain Tor-modules.

In this paper, we compute the projective dimension of the edge ideals of graphs consisting of some cycles and lines which are joint in a common vertex. Moreover, we show that for such graphs, the arithmetical rank equals the projective dimension. As an application, we can compute the arithmetical rank for some homogenous monomial ideals.

Let R be a commutative Noetherian ring, 𝔞 an ideal of R, and M an R-module. We show that, whenever , M is Noetherian if and only if there exists a submodule N of M such that the R-modules M/𝔞 N and are Noetherian. By using this result, we establish Noetherian properties for local cohomology modules in several cases. For instance, we obtain a new version of the Lichtenbaum-Hartshorne Vanishing Theorem.

In this paper we first define a presentation for the generalized Bruck-Reilly ∗-extension of a monoid and then we work on its Gröbner-Shirshov bases.

We study the structure of the set of nilpotent elements in extended semicommutative rings and introduce nil α-semicommutative rings as a generalization. We resolve the structure of nil α-semicommutative rings and obtain various necessary or sufficient conditions for a ring to be nil α-semicommutative, unifying and generalizing a number of known commutative-like conditions in special cases. We also classify which of the standard nilpotence properties on polynomial rings pass to skew polynomial ring. Constructing various examples, we classify how the nil α-semicommutative rings behaves under various ring extensions. Also, we consider the nil-Armendariz condition on a skew polynomial ring.

The authors construct a Wakimoto type realization of toroidal 𝔰𝔩_n+1. The representation constructed in this paper utilizes non-commuting differential operators acting on the tensor product of two polynomial rings in infinitely many commuting variables.

For each linearly normal elliptic curve C ⊂ ℙ³, we determine Galois lines and their arrangement. We prove that the curve C has exactly six V₄-lines. In case j(C) = 1, it has eight Z₄-lines in addition. The V₄-lines form the edges of a tetrahedron. In case j(C) = 1, for each vertex of the tetrahedron, there exist exactly two Z₄-lines passing through it. As a corollary we obtain that each plane quartic curve of genus 1 does not have more than one Galois point.

Starting from a known determinantal representation of outer inverses, we derive their determinantal representation in terms of the inner product in the Euclidean space. We define the double inner product of two miscellaneous tensors of rank 2 in a Riemannian space. The corresponding determinantal representation as well as the general representation of outer inverses in the Riemannian space are derived. A non-zero {2}-inverse X of a given tensor A obeying ρ(X) = s with 1 ≤ s ≤ r = ρ(A) is expressed in terms of the double inner product involving compound tensors with minors of order s, extracted from A and appropriate tensors.

The Lee weight is more appropriate for some practical situations than the Hamming weight as it takes into account of the magnitude of each digit of the word. In this paper, we obtain a sufficient condition over the number of parity check digits for codes correcting simultaneously random and burst errors with Lee weight consideration. This sufficient condition is an extension of the Varshamov-Gilbert-Sacks bound for codes correcting simultaneously random and burst errors with Lee weight constraint.

A well known result of Schur states that if n is a positive integer and a₀, a₁,…,a_n are arbitrary integers with a₀a_n coprime to n!, then the polynomial is irreducible over the field ℚ of rational numbers. In case each a_i = 1, it is known that the Galois group of f_n(x) over ℚ contains A_n, the alternating group on n letters. In this paper, we extend this result to a larger class of polynomials f_n(x) which leads to the construction of trinomials of degree n for each n with Galois group S_n, the symmetric group on n letters.

In this paper, we introduce the concept of 2-absorbing submodules as a generalization of 2-absorbing ideals. Let R be a commutative ring and M an R-module. A proper submodule N of M is called 2-absorbing if whenever a, b ∈ R, m ∈ M and abm ∈ N, then am ∈ N or bm ∈ N or ab ∈ N:_RM. Let N be a 2-absorbing submodule of M. It is shown that N:_RM is a 2-absorbing ideal of R and either Ass_R(M/N) is a totally ordered set or Ass_R(M/N) is the union of two totally ordered sets. Furthermore, it is shown that if M is a finitely generated multiplication module over a Noetherian ring R, and Ass_R(M/N) a totally ordered set, then N is 2-absorbing whenever N:_RM is a 2-absorbing ideal of R. Also, the 2-absorbing avoidance theorem is proved.

In this paper, we introduce the concept of f-ideals and discuss its algebraic properties. In particular, we give a characterization of all the f-ideals of degree 2.

There are a number of identities which, if satisfied by a Bol loop, imply that the loop is actually Moufang. In this paper we show that in a number of cases, the Moufang identity is also forced not by a single identity, but by giving elements a choice of equations to satisfy.

For an integer n ≥ 1, let and S_n be, respectively, the semigroup of partial transformations and the symmetric group on the set X = {1,…,n}. Then S_n is the group of units of . A subsemigroup S of is S_n-normal if for all a ∈ S and g ∈ S_n, g^-1ag ∈ S. In 1976, Symons described the S_n-normal semigroups of full transformations of X. In 1995, Lipscomb and the second author determined the S_n-normal semigroups of partial injective transformations of X. In this paper, we complete the classification by describing all S_n-normal subsemigroups of . As a consequence of the classification theorem, we obtain a characterization of the automorphisms of any S_n-normal subsemigroup of .

Let R be a prime ring, U the Utumi quotient ring of R, C = Z(U) the extended centroid of R, L a non-central Lie ideal of R, and H, G nonzero generalized derivations of R. Suppose that there exists an integer n ≥ 1 such that H(uⁿ)uⁿ + uⁿG(uⁿ) ∈ C for all u ∈ L, then either there exists a ∈ U such that H(x) = xa and G(x) = -ax, or R satisfies the standard identity s₄ and one of the following holds: (i) char(R) = 2; (ii) n is even and there exist a′ ∈ U, α ∈ C and derivations d, δ of R such that H(x) = a′ x + d(x) and G(x) = (α-a′)x + δ(x); (iii) n is even and there exist a′ ∈ U and a derivation δ of R such that H(x)=xa′ and G(x) = -a′ x + δ(x); (iv) n is odd and there exist a′, b′ ∈ U and α, β ∈ C such that H(x) = a′ x + x(β-b′) and G(x) = b′ x+x(α-a′); (v) n is odd and there exist α, β ∈ C and a derivation d of R such that H(x) = α x+d(x) and G(x) = β x + d(x); (vi) n is odd and there exist a′ ∈ U and α ∈ C such that H(x) = xa′ and G(x) = (α - a′)x. As an application of this purely algebraic result, we obtain some range inclusion results of continuous or spectrally bounded generalized derivations H and G on Banach algebras R satisfying the condition H(xⁿ)xⁿ + xⁿG(xⁿ) ∈ rad(R) for all x ∈ R, where rad(R) is the Jacobson radical of R.

We study Specht modules S^(n-2,2) and simple modules D^(n-2,2) for symmetric groups 𝔖_n of degree n over a field of characteristic 2. In particular, we determine the vertices of these modules, and also provide some information on their sources.

Let R be a commutative ring with nonzero identity. In this paper, we study the von Neumann regular elements of R. We also study the idempotent elements, π-regular elements, the von Neumann local elements, and the clean elements of R. Finally, we investigate the subgraphs of the zero-divisor graph Γ(R) of R induced by the above elements.

Given an involution φ : G → G in a group G and a ring R, we study the extensions, not necessarily linear, to an involution ψ : RG → RG in the group ring RG. We investigate the symmetric elements, those α ∈ RG for which ψ(α) = α, and give necessary and sufficient conditions for the set of symmetric elements, (RG)_ψ, to be a subring of RG. This work is a generalization of [6] and references therein where only linear extensions of the group involution are considered.

It is shown that the multiplicative monoids of Brauer's centralizer algebras generated out of the basis are isomorphic to monoids of endomorphisms in categories where an endofunctor is adjoint to itself, and where, moreover, a kind of symmetry involving the self-adjoint functor is satisfied. As in a previous paper, of which this is a companion, it is shown that such a symmetric self-adjunction is found in a category whose arrows are matrices, and the functor adjoint to itself is based on the Kronecker product of matrices.

The aim of this paper is to determine the non-abelian tensor square and Schur multiplier of groups of square-free orders and of groups of orders p²q, pq² and p²qr, where p, q and r are primes and p < q < r.

Let R be a commutative ring with nonzero identity and M be an R-module. Quasi-prime submodules of M and the developed Zariski topology on qSpec(M) are introduced. We also investigate the relationship between algebraic properties of M and topological properties of qSpec(M). Modules whose developed Zariski topology is T₀, irreducible or Noetherian are studied, and several characterizations of such modules are given.

Let R be a commutative ring and M an R-module. In this paper, we study the dual notion of prime submodules (that is, second submodules of M) and investigate the conditions under which the number of maximal second submodules of M is finite. Furthermore, we introduce the concept of coisolated submodules of M and obtain some related characterizations.

Let S be a symbol algebra. The trace form of S is computed and it is shown how this form can be used to determine whether S is a division algebra or not. In addition, the exterior powers of the trace form of S are computed.

A proper subring S of a ring R is said to be maximal if there is no subring of R properly between S and R. If R is a noetherian domain with |R| > 2^ℵ₀, then |Max(R)| ≤ |RgMax(R)|, where RgMax(R) is the set of maximal subrings of R. A useful criterion for the existence of maximal subrings in any ring R is also given. It is observed that if S is a maximal subring of a ring R, then S is artinian if and only if R is artinian and integral over S. Surprisingly, it is shown that any infinite direct product of rings has always maximal subrings. Finally, maximal subrings of zero-dimensional rings are also investigated.

It is shown that if R is a ring with unit element which is not algebraic over the prime subring of R, then R has a maximal subring. It is shown that whenever R ⊆ T are rings such that there exists a maximal subring V of T, which is integrally closed in T and U(R) ⊈ V, then R has a maximal subring. In particular, it is proved that if R is algebraic over ℤ and there exists a natural number n > 1 with n ∈ U(R), then R has a maximal subring. It is shown that if R is an infinite direct product of certain fields, then the maximal ideals M for which R_M (R/M) has maximal subrings are characterized. It is observed that if R is a ring, then either R has a maximal subring or it must be a Hilbert ring. In particular, every reduced ring R with |R|>2^2ℵ₀ or J(R) ≠ 0 has a maximal subring. Finally, the semi-local rings having maximal subrings are fully characterized.

Let M be a normal subgroup of a finite group G such that |G/M| = p^m and |M/Z(M,G)| = pⁿ. In this paper, we characterize all the pairs of groups (M,G) which satisfy the conditions and |[M/Z(M,G),G/Z(M,G)]|≤ p. If M has a complement in G and |M|=pⁿ and |G/M|=p^m, then there always exists a non-negative integer t(M,G) such that . Under some conditions, we determine all the pairs (M,G) with t(M,G)=0, 1 or 2.

Any finitely generated module M over a local ring R is endowed with a complete intersection dimension CI-dim_RM and a Gorenstein dimension G-dim_RM. The Gorenstein dimension can be extended to all modules over the ring R. This paper presents a similar extension for the complete intersection dimension, and mentions the relation between this dimension and the Gorenstein flat dimension. In addition, we show that in the intersection theorem, the flat dimension can be replaced by the complete intersection flat dimension.

Let S be a polynomial ring over a field K and let R be the Stanley-Reisner ring of a matroid complex. In this paper, as a comparison of multiplicity and final Betti number of R over S, the inequality is obtained.

In this paper, we first obtain equivalent conditions for a module over a Prüfer domain and derive properties of Dedekind modules over such a domain. We then obtain equivalent conditions for a finitely generated Dedekind module over an integrally closed ring. Finally, we characterize multiplication Dedekind modules.

Matrix completion problems are an important subclass of problems in matrix theory. An important question in matrix completion problems was posed by Oliveira in 1975, where the author proposed the description of the characteristic polynomial of a partitioned matrix of the form A = [A_i,j], i, j ∈ {1,2} (whose entries are in a field and A_1,1, A_2,2 are square submatrices), when some of the blocks A_i,j are prescribed and the others are unknown. The analysis of this problem gave rise to several subproblems, according to the location of the prescribed submatrices. Many authors have considered this list of subproblems. In this note we provide a new proof of a result obtained by Oliveira inserted in this class of subproblems.

In this research, by using filter regular sequences, we obtain some exact sequences of right or left derived functors of local cohomology modules. Then we use them to gain some conditions under which a right or left derived functor of some special functors over local cohomology modules belongs to a Serre subcategory. These results can conclude some generalizations of previous results in this context or regain some of them.

The structure of groups for which certain sets of commutator subgroups are finite is investigated. In particular, the relation between such groups and groups with finite conjugacy classes of elements is discussed.

We introduce generalized local homology which is in some sense dual to generalized local cohomology, and study some properties of generalized local homology modules for artinian modules, such as the artinianness, noetherianness and the characterization of Width_I(M) by generalized local homology. By using duality, we get back some properties of generalized local cohomology.