Generalized skew derivations with nilpotent values on Lie ideals

Let R be a prime ring and L a noncommutative Lie ideal of R. Suppose that f is a right generalized β-derivation of R associated with a β-derivation δ such that f(x) ⁿ  = 0 for all ${x\in L}$ , where n is a fixed positive integer. Then f = 0.     

On Quadratic Integral Polynomials with only Finitely Many Roots in any Commutative Finite-Dimensional Algebra

The monic quadratic polynomials f with integer coefficients such that each commutative finite-dimensional algebra over a field contains only finitely many roots of f are determined as the polynomials of the form f = X ² + (2m + 1)X + m ² + m, where ${m \in \mathbb{Z}}$ .     

On multiplicative functions which are additive on sums of primes

Let ${f : \mathbb{N} \to \mathbb{C}}$ be a multiplicative function satisfying f(p ₀) ≠ 0 for at least one prime number p ₀, and let k ≥ 2 be an integer. We show that if the function f satisfies f(p ₁ + p ₂ + . . . + p _k ) = f(p ₁) + f(p ₂) + . . . + f(p _k ) for any prime numbers p ₁, p ₂, . . . ,p _k then f must be the identity f(n) = n for each ${n \in \mathbb{N}}$ . This result for k = 2 was established earlier by Spiro, whereas the case k = 3 was recently proved by Fang. In the proof of this result for k ≥ 6 we use a recent result of Tao asserting that every odd number greater than 1 is the sum of at most five primes.     

The McCoy Condition on Skew Polynomial Rings

Based on a theorem of McCoy on commutative rings, Nielsen called a ring R right McCoy if, for any nonzero polynomials f(x), g(x) over R, f(x)g(x) = 0 implies f(x)r = 0 for some 0 ≠ r ? R. In this note, we consider a skew version of these rings, called σ-skew McCoy rings, with respect to a ring endomorphism σ. When σ is the identity endomorphism, this coincides with the notion of a right McCoy ring. Basic properties of σ-skew McCoy rings are observed, and some of the known results on right McCoy rings are obtained as corollaries.     

Traces of Permuting n-Additive Maps and Permuting n-Derivations of Rings

Let R be a prime ring with center Z(R). For a fixed positive integer n, a permuting n-additive map ${\Delta : R^n \to R}$ is known to be permuting n-derivation if ${\Delta(x_1, x_2, \ldots, x_i x'_{i},\ldots, x_n) = \Delta(x_1, x_2, \ldots, x_i, \ldots, x_n)x'_i + x_i \Delta(x_1, x_2, \ldots, x'_i, \ldots, x_n)}$ holds for all ${x_i, x'_i \in R}$ . A mapping ${\delta : R \to R}$ defined by δ(x) = Δ(x, x, . . . ,x) for all ${x \in R}$ is said to be the trace of Δ. In the present paper, we have proved that a ring R is commutative if there exists a permuting n-additive map ${\Delta : R^n \to R}$ such that ${xy + \delta(xy) = yx + \delta(yx), xy- \delta(xy) = yx - \delta(yx), xy - yx = \delta(x) \pm \delta(y)}$ and ${xy + yx = \delta(x) \pm \delta(y)}$ holds for all ${x, y \in R}$ . Further, we have proved that if R is a prime ring with suitable torsion restriction then R is commutative if there exist non-zero permuting n-derivations Δ₁ and Δ₂ from ${R^n \to R}$ such that Δ₁(δ ₂(x), x, . . . ,x) =  0 for all ${x \in R,}$ where δ ₂ is the trace of Δ₂. Finally, it is shown that in a prime ring R of suitable torsion restriction, if ${\Delta_1, \Delta_2 : R^n \longrightarrow R}$ are non-zero permuting n-derivations with traces δ ₁, δ ₂, respectively, and ${B : R^n \longrightarrow R}$ is a permuting n-additive map with trace f such that δ ₁ δ ₂(x) =  f(x) holds for all ${x \in R}$ , then R is commutative.     

Almost f-maps and almost f-rings

In the beginning of the eighties, Buskes and Holland proved that any archimedean almost f-ring is commutative. One decade later, Steinberg showed that if G and H are ? with H archimedean, then any positive bilinear map G × G ${{\begin{array}{ll} B \\ \rightarrow \end{array}}}$ H such that ${x \wedge y = 0}$ implies B(x, y) = 0 is symmetric. At first sight, the Steinberg Theorem might seem to be a considerable generalization of the Buskes and Holland result. It turns out, surprisingly enough, that these two results are equivalent. The main purpose of this paper is to establish this equivalence. A second objective is to apply the aforementioned Steinberg Theorem to prove that if R is an f-ring with a unit element e and S is an archimedean f-ring, then an ?-homomorphism R ${{{\begin{array}{ll} h \\\rightarrow \end{array}}}}$ S is a ring homomorphism if and only if h(e) is idempotent in S. This extends a well-known result by Huijsmans and de Pagter, who obtained the same conclusion for semiprime f-algebras.     

A Note on Comaximal Graph of Non-commutative Rings

Let R be a ring with unity. The graph Γ(R) is a graph with vertices as elements of R, where two distinct vertices a and b are adjacent if and only if Ra?+?Rb?=?R. Let Γ₂(R) be the subgraph of Γ(R) induced by the non-unit elements of R. Let R be a commutative ring with unity and let J(R) denote the Jacobson radical of R. If R is not a local ring, then it was proved that:

1. If $\Gamma_2(R)\backslash J(R)$ is a complete n-partite graph, then n?=?2.
2. If there exists a vertex of $\Gamma_2(R)\backslash J(R)$ which is adjacent to every vertex, then R????₂×F, where F is a field.

In this note we generalize the above results to non-commutative rings and characterize all non-local ring R (not necessarily commutative) whose $\Gamma_2(R)\backslash J(R)$ is a complete n-partite graph.     

On the Strong (A)-Rings of Mahdou and Hassani

A (commutative unital) ring R with only finitely many minimal prime ideals (for instance, a Noetherian ring) is reduced and a strong (A)-ring if and only if R is an integral domain. Thus, the smallest reduced ring which has Property A but is not a strong (A)-ring is ${\mathbb{Z}_{2} \times \mathbb{Z}_{2}}$ . A Noetherian ring R is a strong (A)-ring if and only if Ass _R (R) has a unique maximal element.     

On certain functional equation related to two-sided centralizers

Let m ≥ 0, n ≥ 0 be fixed integers with m + n ≠ 0 and let R be a prime ring with char(R) = 0 or m + n + 1 ≤ char(R) ≠ 2. Suppose that there exists an additive mapping T : R → R satisfying the relation 2T(x ^m+n+1) = x ^m T(x) x ⁿ  + x ⁿ T(x)x ^m for all ${x\in R}$ . In this case T is a two-sided centralizer.     

On locally bounded above solutions of an equation of the Goła̧b–Schinzel type

We characterize solutions ${f, g : \mathbb{R} \to \mathbb{R}}$ of the functional equation f(x + g(x)y) = f(x)f(y) under the assumption that f is locally bounded above at each point ${x \in \mathbb{R}}$ . Our result refers to Go?a?b and Schinzel (Publ Math Debr 6:113–125, 1959) and Wo?od?ko (Aequationes Math 2:12–29, 1968).     

Roman Domination Dot-critical Graphs

A Roman dominating function on a graph G is a function f : V(G) → {0, 1, 2} satisfying the condition that every vertex u for which f (u) = 0 is adjacent to at least one vertex v for which f (v) = 2. The weight of a Roman dominating function is the value ${f(V(G))=\sum_{u \in V(G)}f(u)}$ . The Roman domination number, γ _R (G), of G is the minimum weight of a Roman dominating function on G. In this paper, we study graphs for which contracting any edge decreases the Roman domination number.     

On the Discriminant Scheme of Homogeneous Polynomials

In this paper, the discriminant scheme of homogeneous polynomials is studied in two particular cases: the case of a single homogeneous polynomial and the case of a collection of n ? 1 homogeneous polynomials in \({n\geqslant 2}\) variables. In both situations, a normalized discriminant polynomial is defined over an arbitrary commutative ring of coefficients by means of the resultant theory. An extensive formalism for this discriminant is then developed, including many new properties and computational rules. Finally, it is shown that this discriminant polynomial is faithful to the geometry: it is a defining equation of the discriminant scheme over a general coefficient ring k, typically a domain, if \({2\neq 0}\) in k. The case where 2 = 0 in k is also analyzed in detail.     

A Quadratic Differential Identity with Generalized Derivations on Multilinear Polynomials in Prime Rings

Let R be a prime ring of characteristic different from 2, with right Utumi quotient ring U and extended centroid C, and let ${f(x_1, \ldots, x_n)}$ be a multilinear polynomial over C, not central valued on R. Suppose that d is a derivation of R and G is a generalized derivation of R such that $$G(f(r_1, \ldots, r_n))d(f(r_1, \ldots, r_n)) + d(f(r_1, \ldots, r_n))G(f(r_1, \ldots, r_n)) = 0$$ for all ${r_1, \ldots, r_n \in R}$ . Then either d =  0 or G =  0, unless when d is an inner derivation of R, there exists ${\lambda \in C}$ such that G(x) =  λ x, for all ${x \in R}$ and ${f(x_1, \ldots, x_n)^2}$ is central valued on R.     

Power automorphisms of finite p-groups

Abstract

In this paper, the definitions of quasi-orthogonal idempotent sequences and Iq-dimensions of a ring R are given. The relations between Iq-dimensions and block decomposition numbers of a ring are discussed. As a generalization of monic polynomials, the concept of quasi-monic polynomials over a ring is introduced. It is shown that, for a quasi-monic polynomial over a ring, the division algorithm holds. Suslin Lemma and Horrocks' Theorem are extended to the setting of quasi-monic polynomials. For a commutative ring R, if f(x) is a quasi-monic polynomial in R[x], then GD(R) = GD(R[x]/f(x)) is proved, where GD(R) denotes the global dimension of R.     

On the canonical ring of curves and surfaces

Let C be a curve (possibly non reduced or reducible) lying on a smooth algebraic surface. We show that the canonical ring ${ R(C, \omega_C)=\bigoplus_{k\geq 0} H^0(C, {\omega_C}^{\otimes k})}$ is generated in degree 1 if C is numerically four-connected, not hyperelliptic and even (i.e. with ω _C of even degree on every component). As a corollary we show that on a smooth algebraic surface of general type with p _g (S) ≥ 1 and q(S) = 0 the canonical ring R(S, K _S ) is generated in degree ≤  3 if there exists a curve ${C \in |K_S|}$ numerically three-connected and not hyperelliptic.     

Around a problem of Nicole Brillouët–Belluot

We determine nontrivial intervals \({I \subset(0,+\infty)}\) , numbers \({\alpha\in\mathbb R}\) and continuous bijections \({f \colon I \to I}\) such that f(x)f ^?1(x) = x ^α for every \({x\in I}\) .     

Varieties of strange plane curves

Abstract

Let A be a commutative ring with identity, let X, Y be indeterminates and let F(X,Y), G(X, Y) ∈ A[X, Y] be homogeneous. Then the pair F(X, Y), G(X, Y) is said to be radical preserving with respect to A if Rad((F(x, y), G(x, y))R) = Rad((x,y)R) for each A-algebra R and each pair of elements x, y in R. It is shown that infinite sequences of pairwise radical preserving polynomials can be obtained by homogenizing cyclotomic polynomials, and that under suitable conditions on a ?-graded ring A these can be used to produce an infinite set of homogeneous prime ideals between two given homogeneous prime ideals P ? Q of A such that ht(Q/P) = 2.     

The existence of maximal (q 2, 2)-arcs in projective Hjelmslev planes over chain rings of length 2 and odd prime characteristic

We prove that (q ², 2)-arcs exist in the projective Hjelmslev plane PHG(2, R) over a chain ring R of length 2, order |R| = q ² and prime characteristic. For odd prime characteristic, our construction solves the maximal arc problem. For characteristic 2, an extension of the above construction yields the lower bound q ² + 2 on the maximum size of a 2-arc in PHG(2, R). Translating the arcs into codes, we get linear [q ³, 6, q ³ ?q ² ?q] codes over ${\mathbb {F}_q}$ for every prime power q > 1 and linear [q ³ + q, 6,q ³ ?q ² ?1] codes over ${\mathbb {F}_q}$ for the special case q = 2 ^r . Furthermore, we construct 2-arcs of size (q + 1)²/4 in the planes PHG(2, R) over Galois rings R of length 2 and odd characteristic p ².     

Fast Growing Solutions of Linear Differential Equations in the Unit Disc

For $n \in \mathbb{N}$ , the n-order of an analytic function f in the unit disc D is defined by $$\sigma _{{{M,n}}} (f) = {\mathop {\lim \sup }\limits_{r \to 1^{ - } } }\frac{{\log ^{ + }_{{n + 1}} M(r,f)}} {{ - \log (1 - r)}},$$ where log⁺ x  =  max{log x, 0}, log ⁺ ₁ x  =  log ⁺ x, log ⁺ _n+1 x  =  log ⁺ log ⁺ _n x, and M(r, f) is the maximum modulus of f on the circle of radius r centered at the origin. It is shown, for example, that the solutions f of the complex linear differential equation $$f^{{(k)}} + a_{{k - 1}} (z)f^{{(k - 1)}} + \cdots + a_{1} (z)f^{\prime} + a_{0} (z)f = 0,\quad \quad \quad (\dag)$$ where the coefficients are analytic in D, satisfy σ _M,n+1(f)  ≤  α if and only if σ _M,n (a _j )  ≤  α for all j  =  0, ..., k ? 1. Moreover, if q ∈{0, ..., k ? 1} is the largest index for which $\sigma _{M,n} ( a_{q}) = {\mathop {\max }\limits_{0 \leq j \leq k - 1} }{\left\{ {\sigma _{{M,n}} {\left( {a_{j} } \right)}} \right\}}$ , then there are at least k ? q linearly independent solutions f of ( $\dag$ ) such that σ _M,n+1(f) = σ _M,n (a _q ). Some refinements of these results in terms of the n-type of an analytic function in D are also given.     

Finite determinacy and Whitney equisingularity of map germs from {\mathbb{C}^n} to {\mathbb{C}^{2n-1}}

We show that a holomorphic map germ ${f : (\mathbb{C}^n,0)\to(\mathbb{C}^{2n-1},0)}$ is finitely determined if and only if the double point scheme D(f) is a reduced curve. If n ≥ 3, we have that μ(D ²(f)) = 2μ(D ²(f)/S ₂)+C(f)?1, where D ²(f) is the lifting of the double point curve in ${(\mathbb{C}^n\times \mathbb{C}^n,0)}$ , μ(X) denotes the Milnor number of X and C(f) is the number of cross-caps that appear in a stable deformation of f. Moreover, we consider an unfolding F(t, x) = (t, f _t (x)) of f and show that if F is μ-constant, then it is excellent in the sense of Gaffney. Finally, we find a minimal set of invariants whose constancy in the family f _t is equivalent to the Whitney equisingularity of F. We also give an example of an unfolding which is topologically trivial, but it is not Whitney equisingular.