Journal of Algebra and Its Applications

Analogous to left p.q.-Baer property of a ring [G. F. Birkenmeier, J. Y. Kim and J. K. Park, Principally quasi-Baer rings, Comm. Algebra29 (2001) 639–660], we say a right R-module M is endo-principallyquasi-Baer (or simply, endo-p.q.-Baer) if for every m ∈ M, l_S(Sm) = Se for some e² = e ∈ S = End_R(M). It is shown that every direct summand of an endo-p.q.-Baer module inherits the property that any projective (free) module over a left p.q.-Baer ring is an endo-p.q.-Baer module. In particular, the endomorphism ring of every infinitely generated free right R-module is a left (or right) p.q.-Baer ring if and only if R is quasi-Baer. Furthermore, every principally right ℱℐ-extending right ℱℐ-𝒦-nonsingular ring is left p.q.-Baer and every left p.q.-Baer right ℱℐ-𝒦-cononsingular ring is principally right ℱℐ-extending.

The induced topology on the points of the Connes–Consani "arithmetic site," which are the finite adèle classes, is family coarse. In this note we define another topology on this set having properties one might expect of the mythical object .

In this paper, the alternating groups of degree p are characterized by their order and one conjugacy class length, where p is a prime number. This work implies that Thompson's conjecture holds for the alternating groups of degree p.

In this paper we revisit that ideal lattices and congruence lattices are preserved by Morita equivalence of semirings which is originally obtained implicitly by Katsov and his co-authors. This is then used to obtain some Morita invariants for semirings.

We study a class of down–up algebras 𝒜(α, β, ϕ) defined over a polynomial base ring 𝕂[t₁,…,t_n] and establish several analogous results. We first construct a 𝕂-basis for the algebra 𝒜(α, β, ϕ). As an application, we completely determine the center of 𝒜(α, β, ϕ) when char 𝕂 = 0, and prove that the Gelfand–Kirillov dimension of 𝒜(α, β, ϕ) is n + 3. Then, we prove that 𝒜(α, β, ϕ) is a noetherian domain if and only if β ≠ 0, and 𝒜(α, β, ϕ) is Auslander-regular when β ≠ 0. We show that the global dimension of 𝒜(α, β, ϕ) is n + 3, and 𝒜(α, β, ϕ) is a prime ring except when α = β = ϕ = 0. Finally, we obtain some results on the Krull dimensions, isomorphisms and automorphisms of 𝒜(α, β, ϕ).

The paper provides two constructions of compressed sensing matrices using the subspaces of symplectic space and singular symplectic space over finite fields. Then we compare the matrices constructed in this paper with the matrix constructed by DeVore, and compare the two matrices based on symplectic geometry and singular symplectic geometry over finite fields.

Let G be a group and Z_j(G), for j ≥ 0, be the jth term in the upper central series of G. We prove that if Out^c(G/Z_j(G)), the group of outer class-preserving automorphisms of G/Z_j(G), is nilpotent of class k, then Out^c(G) is nilpotent of class at most j + k. Moreover, if Out^c(G/Z_j(G)) is a trivial group, then Out^c(G) is nilpotent of class at most j. As an application we prove that if γ_i(G)/γ_i(G) ∩ Z_j(G) is cyclic then Out^c(G) is nilpotent of class at most i + j, where γ_i(G), for i ≥ 1, denotes the ith term in the lower central series of G. This extends an earlier work of the author, where this assertion was proved for j = 0. We also improve bound on the nilpotency class of Out^c(G) for some classes of nilpotent groups G.

Let p₁ ≡ p₂ ≡ -q ≡ 1 (mod 4) be primes such that and . Put and d = p₁p₂q, then the bicyclic biquadratic field has an elementary Abelian 2-class group of rank 3. In this paper we determine the nilpotency class, the coclass, the generators and the structure of the non-Abelian Galois group of the second Hilbert 2-class field of 𝕂, we study the 2-class field tower of 𝕂, and we study the capitulation problem of the 2-classes of 𝕂 in its fourteen abelian unramified extensions of relative degrees two and four.

Ring extensions are a well-studied topic in ring theory. In this paper, we study the structure of the Gauss extension of a Galois ring. We determine the structures of the extension ring and its unit group.

Let 𝔤 be a finite-dimensional Lie algebra over a field of characteristic 0, with solvable radical 𝔯 and nilpotent radical 𝔫 = [𝔤, 𝔯]. Given a finite-dimensional 𝔤-module U, its nilpotency series 0 ⊂ U⁽¹⁾ ⊂ ⋯ ⊂ U^(m) = U is defined so that U⁽¹⁾ is the 0-weight space of 𝔫 in U, U⁽²⁾/U⁽¹⁾ is the 0-weight space of 𝔫 in U/U⁽¹⁾, and so on. We say that U is linked if each factor of its nilpotency series is a uniserial 𝔤/𝔫-module, i.e. its 𝔤/𝔫-submodules form a chain. Every uniserial 𝔤-module is linked, every linked 𝔤-module is indecomposable with irreducible socle, and both converses fail. In this paper, we classify all linked 𝔤-modules when 𝔤 = 〈x〉 ⋉ 𝔞 and ad x acts diagonalizably on the abelian Lie algebra 𝔞. Moreover, we identify and classify all uniserial 𝔤-modules amongst them.

In this paper, we study the class of rings in which every pure ideal is projective. We refer to rings with this property as PIP-rings. Some properties and examples of PIP-rings are given. When R is a PIP-ring, some new homological dimensions for complexes are given. As applications, we give some new characterizations of von Neumann regular rings, F-rings and semisimple Artinian rings.

The purpose of this paper is to examine the freeness of nonzero reflexive modules M and N over a regular local ring R, under the condition is zero.

It is known that the integral cohomology algebra of any smooth compact toric variety X_Σ associated to a complete regular fan Σ is isomorphic to the Stanley–Reisner algebra ℤ[Σ] modulo the ideal J_Σ generated by linear relations determined by Σ. The aim of this paper is to show how to determine the integral cohomology algebra of a toric variety (in particular, a projective toric variety) associated to a certain simplicial fan. As a consequence, we confirm our expectation that for a certain simplicial fan the integral cohomology algebra is also given by the same formula as in a complete regular fan.

An element of a group G is called semi-rational if there is a natural number m such that each generator of 〈x〉 belongs to the conjugacy class of G containing x or x^m. If all elements of G are semi-rational, then G is called a semi-rational group. In this paper, we study semi-rational Frobenius groups G and obtain results concerning effect of semi-rationality property on the kernel and complement of G. In particular, we show that ∣π(G)∣ ≤ 5 which answers Problem 2 in [D. Chillag and S. Dolfi, Semi-rational solvable groups, J. Group Theory13(4) (2010) 535–548] for semi-rational Frobenius groups.

We introduce quasi-Poisson cohomology groups for a Poisson algebra, which can be computed by its quasi-Poisson complex. Moreover, there exists a Grothendieck spectral sequence relating quasi-Poisson cohomology to Hochschild cohomology and Lie algebra cohomology.

I construct a Koszul algebra A and a finitely generated graded A-module M that together form a counterexample to a recently published claim. M is generated in degree 0 and has a pure resolution, and the graded Jacobson radical of the Yoneda algebra of A does not annihilate the Ext module of M, but nonetheless M is not a Koszul module.

D. Puder defined the primitivity rank of elements of free groups [Primitive words, free factors and measure preservation, Israel J. Math.201(1) (2014) 25–73], we give a similar definition for free algebras of Schreier varieties and prove properties of a primitivity rank using the properties of the almost primitive elements.

Let H be a subgroup of Sym_n, the symmetric group of degree n. For a fixed integer l ≥ 2, the group G presented with generators x₁, x₂,…,x_n and with relations x_i1x_i2⋯x_il = x_σ(i1) x_σ(i2)⋯x_σ(il), where σ runs through H, is considered. It is shown that G has a free subgroup of finite index. For a field K, properties of the algebra K[G] are derived. In particular, the Jacobson radical 𝒥(K[G]) is always nilpotent, and in many cases the algebra K[G] is semiprimitive. Results on the growth and the Gelfand–Kirillov dimension of K[G] are given. Further properties of the semigroup S and the semigroup algebra K[S] with the same presentation are obtained, in case S is cancellative. The Jacobson radical is nilpotent in this case as well, and sufficient conditions for the algebra to be semiprimitive are given.

Let q be a nonzero complex number. In this paper, we completely determine the derivation algebras and the automorphism groups of the Block type Lie algebras ℬ(q).