Journal of Algebra and Its Applications

Let be the full transformation semigroup on a nonempty set X and E be an equivalence relation on X. Then

Let A be an f-ring with identity u and B be an archimedean f-ring. For every idempotent element w in B, let denote the set of all positive group homomorphisms ℌ : A → B with ℌ(u) = w. We prove that is a ring homomorphism if and only if ℌ is an extreme point of . As a consequence, we derive a characterization of ring homomorphisms in in terms of a Gelfand-type transform. Moreover, we show that ring homomorphisms in are, up to multiplicative constants, all the basic elements of the ℓ-group of all bounded group homomorphisms from A into ℝ.

Let D be a division ring with center F. An element of the form xyx^-1y^-1 ∈ D is called a multiplicative commutator. Let T(D) be the vector space over F generated by all multiplicative commutators in D. In this paper it is shown that if D is algebraic over F and Char(D) = 0, then D = T(D). We conjecture that it is true in general. Among other results it is shown that in characteristic zero if T(D) is algebraic over F, then D is algebraic over F.

Let K be a field of odd characteristic p, and let G be the direct product of a finite p-group P ≠ 1 and a Hamiltonian 2-group. We show that the set of symmetric elements (KG)_* of the group algebra KG with respect to the involution of KG which inverts all elements of G, satisfies all Lie commutator identities of degree t(P) or more, where t(P) denotes the nilpotency index of the augmentation ideal of the group algebra KP. In addition, if P is powerful, then (KG)_* satisfies no Lie commutator identity of degree less than t(P). Applying this result we get that (KG)_* is Lie nilpotent and Lie solvable, and its Lie nilpotency index and Lie derived length are not greater than t(P) and ⌈log₂ t(P)⌉, respectively, and these bounds are attained whenever P is a powerful group. The corresponding result on the set of symmetric units of KG is also obtained.

It is known that the degree of nilpotency of a nilpotent derivation on a prime ring including the ring of n × n matrices must be an odd number. In this article we introduce subrings of the ring of of n × n matrices that admit derivations with an even degree of nilpotency.

Let F be a finite field and let Sp_2ν(F) be the symplectic group over F. If Sp_2ν(F) acts on the F-vector space F^2ν, then it can induce an action on the vector space F^2ν ⊕ F^2ν, defined by (x, y)^A = (xA, yA), ∀ x, y ∈ F^2ν, A ∈ Sp_2ν(F). If K is a field with char K ≠ char F, then Sp_2ν(F) also acts on the group algebra K[F^2ν ⊕ F^2ν]. In this paper, we determine the structures of Sp_2ν(F)-stable ideals of the group algebra K[F^2ν ⊕ F^2ν] by augmentation ideals, and describe the relations between the invariant ideals of K[F^2ν] and the vector invariant ideals of K[F^2ν ⊕ F^2ν].

Suppose G is the zero-divisor graph of some commutative ring with 1. When G has four or more vertices, a method is presented to find a specific commutative ring R with 1 such that Γ(R) ≅ G. Furthermore, this ring R can be written as R ≅ R₁ × R₂ × ⋯ × R_n, where each R_i is local and this representation of R is unique up to factors R_i with isomorphic zero-divisor graphs. It is also shown that for graphs on four or more vertices, no local ring has the same zero-divisor graph as a non-local ring and no reduced ring has the same zero-divisor graph as a non-reduced ring.

In this paper we classify non-commutative quadrics and study their homological properties. In fact we find all non-commutative algebras of degree 2 up to isomorphism and we study these algebras via their homomorphic images onto the polynomial algebra k[x,y] as well as the Ext¹(k(p), k(q))-groups, where k(p) and k(q) are one-dimensional simple modules. Moreover some general results on simple finite-dimensional modules are obtained. Some of these results are applied to the special cases of non-commutative quadrics.

Biracks are algebraic structures related to knots and links. We define a new enhancement of the birack counting invariant for oriented knots and links via algebraic structures called birack dynamical cocycles. The new invariants can also be understood in terms of partitions of the set of birack labelings of a link diagram determined by a homomorphism p : X → Y between finite labeling biracks. We provide examples to show that the new invariant is stronger than the unenhanced birack counting invariant and examine connections with other knot and link invariants.

We investigate purities determined by classes of finitely presented modules including the correspondence between purities for left and right modules. We show some cases where purities determined by matrices of given sizes are different. Then we consider purities over finite-dimensional algebras, giving a general description of the relative pure-injectives which we make completely explicit in the case of tame hereditary algebras.

A derivation of an associative ring R is an additive map satisfying T(xy) = T(x)y + xT(y) for all x, y in R. We study rings with a derivation T satisfying Herstein's condition [T(R), T(R)] = 0. (The commutator [u, v] is defined by: [u, v] = uv - vu.) This work studies the structure of the ideal I generated by T(R). We show that I³ is in the center of R, and we show that R has an ideal K which is contained in the kernel of T, K² = 0, and [T(R/K), R/K] generates a trivial ideal of R/K.

In this paper, finite p-groups all of whose non-normal abelian subgroups are cyclic are classified up to isomorphism.

In this paper, we find an upper bound for the exponent of the Schur multiplier of a pair (G, N) of finite p-groups, when N admits a complement in G. As a consequence, we show that the exponent of the Schur multiplier of a pair (G, N) divides exp(N) if (G, N) is a pair of finite p-groups of class at most p – 1. We also prove that if N is powerfully embedded in G, then the exponent of the Schur multiplier of a pair (G, N) divides exp(N).

We decompose the -module V(Λ₀) ⊗ V(Λ₀) and give generating function identities for the outer multiplicities. In the process we discover an infinite family of partition identities, which are seemingly new even in the n = 3 case.

Given a compact linear Lie group G, we form a natural expansion of the theory of the reals where G and the graph of a logarithm function on G live. We prove its effective model-completeness and decidability modulo a suitable variant of Schanuel's Conjecture.

Let R be a commutative ring with unity. The cozero-divisor graph of R denoted by Γ'(R) is a graph with the vertex set W*(R), where W*(R) is the set of all nonzero and non-unit elements of R, and two distinct vertices a and b are adjacent if and only if a ∉ Rb and b ∉ Ra, where Rc is the ideal generated by the element c in R. Let α(Γ'(R)) and γ(Γ'(R)) denote the independence number and the domination number of Γ'(R), respectively. In this paper, we prove that if α(Γ'(R)) is finite, then R is Artinian if and only if R is Noetherian. Also, we prove that if α(Γ'(R)) is finite, then R/P is finite, for every prime ideal P. Moreover, we prove that if R is a Noetherian ring, γ(Γ'(R)) is finite and Γ'(R) has at least one isolated vertex, then J(R) = N(R). We show that if R is a commutative Noetherian local ring, γ(Γ'(R)) is finite and Γ'(R) has at least one isolated vertex, then R is a finite ring. Among other results, we prove that if R is a commutative ring and the maximum degree of Γ'(R) is finite and positive, then R is a finite ring.

A ring R is said to be right serial, if it is a direct sum of right ideals which are uniserial. A ring that is right serial need not be left serial. Right artinian, right serial ring naturally arise in the study of artinian rings satisfying certain conditions. For example, if an artinian ring R is such that all finitely generated indecomposable right R-modules are uniform or all finitely generated indecomposable left R-modules are local, then R is right serial. Such rings have been studied by many authors including Ivanov, Singh and Bleehed, and Tachikawa. In this paper, a universal construction of a class of indecomposable, non-local, basic, right artinian, right serial rings is given. The construction depends on a right artinian, right serial ring generating system X, which gives rise to a tensor ring T(L). It is proved that any basic right artinian, right serial ring is a homomorphic image of one such T(L).

Let be a ring. We say that is zero product determined if for every additive group and every bi-additive map the following holds: if ϕ(a, b) = 0 whenever ab = 0, then there exists an additive map such that ϕ(a, b) = T(ab) for all . In this paper, first we study the properties of zero product determined rings and show that semi-commutative and non-commutative rings are not zero product determined. Then, we will examine whether the rings with a nontrivial idempotent are zero product determined. As applications of the above results, we prove that simple rings with a nontrivial idempotent, full matrix rings and some classes of operator algebras are zero product determined rings and discuss whether triangular rings and upper triangular matrix rings are zero product determined.

In this note, we have given the center Z(V_*(𝔽_2^mM₁₆)) of unitary units subgroup V_*(𝔽_2^mM₁₆) of group algebra 𝔽_2^mM₁₆, where M₁₆ = 〈x, y | x⁸ = y² = 1, xy = yx⁵〉 is the Modular group of order 16 and 𝔽_2^m is any finite field of characteristic 2, with 2^m elements. The structure of the unitary unit subgroup V_*(𝔽_2^mM₁₆) of the group algebra 𝔽_2^mM₁₆, is also described, see Theorem 3.1.

Let ℨ be a complete set of Sylow subgroups of a finite group G, that is, for each prime p dividing the order of G, ℨ contains exactly one and only one Sylow p-subgroup of G, say G_p. Let C be a nonempty subset of G. A subgroup H of G is said to be C-ℨ-permutable (conjugate-ℨ-permutable) subgroup of G if there exists some x ∈ C such that H^xG_p = G_pH^x, for all G_p ∈ ℨ. We investigate the structure of the finite group G under the assumption that certain subgroups of prime power orders of G are C-ℨ-permutable (conjugate-ℨ-permutable) subgroups of G. Our results improve and generalize several results in the literature.