Additive Endomorphisms Research Articles

Equivariant endomorphisms of convex functions

Characterizations of all continuous, additive and GL(n)-equivariant endomorphisms of the space of convex functions on a Euclidean space Rn, of the subspace of convex functions that are finite in a neighborhood of the origin, and of finite convex functions are established. Moreover, all continuous, additive, monotone endomorphisms of the same spaces, which are equivariant with respect to rotations and dilations, are characterized. Finally, all continuous, additive endomorphisms of the space of finite convex functions of one variable are characterized.

NEAR-RING PRIMA DAN SEMIPRIMA YANG DILENGKAPI DENGAN DERIVATIF

Let 𝑁 be a semiprime near-ring with 𝑑 derivations of 𝑁. Derivations are referred to group additive endomorphism with multiplication operating of 𝑑(𝑥. 𝑦)= 𝑥𝑑(𝑦)+ 𝑑(𝑥)𝑦 = 0� for each 𝑥, 𝑦 ∈ 𝑁. This paper gives sufficient conditions on a subset near-ring order derivation of each of its members is equal to 0. Let N be a semiprime near-ring and A�N such that 0 ∈ 𝐴,𝐴. 𝑁 ⊆ 𝐴 and d derivation of N. The purpose of this paper is to prove that if d acts as a homomorphism on A or as an anti-homomorphism on then d(A) = 0

On the arithmetic of the BC-system

For each prime p and each embedding \sigma of the multiplicative group of an algebraic closure of \mathbb{F}_p as complex roots of unity, we construct a p -adic indecomposable representation \pi_\sigma of the integral BC-system as additive endomorphisms of the big Witt ring of \bar{\mathbb{F}}_p . The obtained representations are the p -adic analogues of the complex, extremal KMS _\infty states of the BC-system. The role of the Riemann zeta function, as partition function of the BC-system over \mathbb{C} is replaced, in the p -adic case, by the p -adic L -functions and the polylogarithms whose values at roots of unity encode the KMS states. We use Iwasawa theory to extend the KMS theory to a covering of the completion \mathbb{C}_p of an algebraic closure of \mathbb{Q}_p . We show that our previous work on the hyperring structure of the adèle class space, combines with p -adic analysis to refine the space of valuations on the cyclotomic extension of \mathbb{Q} as a noncommutative space intimately related to the integral BC-system and whose arithmetic geometry comes close to fulfill the expectations of the “arithmetic site”. Finally, we explain how the integral BC-system appears naturally also in de Smit and Lenstra construction of the standard model of \bar{\mathbb{F}}_p which singles out the subsystem associated to the \hat{\mathbb{Z}} -extension of \mathbb{Q} .

Extremal case in Marcus–Oliveira conjecture and beyond

For A, C ∈ M n the C-determinantal range of A is the following set on the complex plane ▵ C (A) = {det(A − UCU*): UU* = I n }. For normal matrices A and C with eigenvalues α1, … , α n and γ1, … , γ n , respectively, Marcus [M. Marcus, Derivations, Plücker relations and the numerical range, Indiana Univ. Math. J. 22 (1973), pp. 1137–1149] and Oliveira [G.N. de Oliveira, Normal matrices (research problem), Linear Multilinear Algebra 12 (1982), pp. 153–154] conjectured that ▵ C (A) is a subset of the convex hull of the points , σ ∈ S n , where S n is the symmetric group of degree n. We investigate the extremal set of matrices for which the equality holds in the Marcus–Oliveira conjecture. We illustrate the use of the obtained results by two different applications. The first one deals with the equality case between the radius of ▵ C (A) and the radius of the convex hull of the points z σ, σ ∈ S n . The second one is the characterization of additive Frobenius endomorphisms for the determinantal range or radius on the space M n and on its real subspace of Hermitian matrices.

On definability of a periodic EndE+-group by its endomorphism group

Let A be a class of Abelian groups, A ∈ A, and End(A) be the additive endomorphism group of the group A. The group A is said to be defined by its endomorphism group in the class {ie208-01} if for every group B ∈ B such that End(B) ≅ End(A) the isomorphism B ≅ A holds. The paper considers the problem of definability of a periodic Abelian group A such that End-End(A) ≅ End(A). The classes of periodical Abelian groups, of divisible Abelian groups, of reduced Abelian groups, of nonreduced Abelian groups, and of all Abelian groups are investigated in this paper.

Rings with Zassenhaus Families of Ideals

Let R be a ring with identity. We call a family ℱ of left ideals of R a Zassenhaus family if the only additive endomorphisms of R that leave all members of ℱ invariant are the left multiplications by elements of R. Moreover, if R is torsion-free and there is some left R-module M such that R ⊆ M ⊆ R⊗ℤℚ and End ℤ(M) = R we call R a “Zassenhaus ring”. It is well known that all Zassenhaus rings have Zassenhaus families. We will give examples to show that the converse does not hold even for torsion-free rings of finite rank.

Derivations and Skew Polynomial Rings

Let R be a prime ring and d a derivation of R. In the ring of additive endomorphisms of the Abelian group (R, +), let S be the subring generated by a L d m , where a ∈ R and m ≥ 0 and where a L :x ∈ R ↦ ax ∈ R for a ∈ R. We compute the prime radical and minimal prime ideals of S via the skew polynomial ring R[x; d] by the surjective ring homomorphism We compute explicitly the kernel 𝒜 of ϕ, the prime radical 𝒫 over 𝒜 and minimal prime ideals over 𝒜 (Theorem 2). We obtain a necessary and sufficient condition for S to be simple, prime or semiprime (Corollary 3). As an application, let d be nilpotent. We show that the d-extension of R defined in Grzeszczuk (1992) is canonically isomorphic to the quotient ring of S modulo its prime radical (Corollary 14).

Rings whose additive endomorphisms aren-multiplicative, II

A ringR is called anAE n -ring,n≥2 a positive integer, if every endomorphism ϕ of additive group ofR satisfies ϕ(a 1 a 2...a n )=ϕ(a 1)ϕ(a 2)...ϕ(a n ) for alla 1,...,a n eR. Several results concerning the structure ofAE n -rings are obtained in this note, including an (incomplete) description ofAE n -ringsR satisfyingR t R n−1 ≠0, whereR t is the torsion ideal inR.

Rings whose additive endomorphisms are ring endomorphisms

A ring R is said to be an AE-ring if every endomorphism of its additive group R+ is a ring endomorphism. Clearly, the zero ring on any abelian group is an AE-ring. In a recent article, Birkenmeier and Heatherly characterised the so-called standard AE-lings, that is, the non-trivial AE-rings whose maximal 2-subgroup is a direct summand. The present article demonstrates the existence of non-standard AE-rings. Four questions posed by Birkenmeier and Heatherly are answered in the negative.

Rings whose additive endomorphisms are multiplicative

Rings whose additive endomorphisms are multiplicative

On rings whose additive endomorphisms are multiplicative

On rings whose additive endomorphisms are multiplicative

Rings whose additive endomorphisms are ring endomorphisms

A ring R is said to be an AE-ring if every additive endomorphism is a ring endomorphism. In this paper further steps are made toward solving Sullivan's Problem of characterising these rings. The classification of AE-rings with. R3 ≠ 0 is completed. Complete characterisations are given for AE-rings which are either: (i) subdirectly irreducible, (ii) algebras over fields, or (iii) additively indecomposable. Substantial progress is made in classifying AE-rings which are mixed – the last open case – by imposing various finiteness conditions (chain conditions on special ideals, height restricting conditions). Several open questions are posed.

An expansion of F̃p

Let K be a field of characteristic p. The map τ(X) = Xp − X is an additive endomorphism of K, with kernel Fp. The Galois extensions of K of order p are obtained by adjoining to K solutions to equations of the form Xp − X = a for some a in K. These extensions are called the Artin-Schreier extensions of K and have a cyclic Galois group.The study of Artin-Schreier extensions is very important for studying fields of characteristic p, in particular for studying valued fields of the form K((t)). An attempt at getting quantifier elimination for those fields would necessitate the adjunction to the language of fields of a cross-section for the function τ, i.e. a function σ such that τ ∘ σ is the identity on the image of τ. When K = Fp, such a cross-section is in fact definable in K((t)): it associates to τ(x) the element of {x, x + 1, …, x + p – 1} whose constant term is 0 (see [2]). When K is infinite, such a cross-section is usually not definable.The results presented in this paper originate from a question of L. van den Dries: is there a natural way of defining a cross-section σ for τ on F̃p, and is the theory of (F̃p, σ) decidable? (F̃p is the algebraic closure of Fp.)

Rings whose additive endomorphisms are N-multiplicative

Sullivan's problem of describing rings, all of whose additive endomorphisms are multiplicative, is generalised to the study of rings R satisfying ϕ(a1 … an) = ϕ(a1)…ϕ(an) for every additive endomorphism ϕ of R, and all a1,…,an ∈ R, with n > 1 a fixed positive integer. It is shown that such rings possess a bounded (finite) ideal A such that [R/A]n = 0 ([R/A]2n−1 = 0). More generally, if f(X1, …, Xt) is a homogeneous polynomial with integer coefficients, of degree > 1, and if a ring R satisfies ϕ[f(a1, …, at)] = f[ϕ(a1), …, ϕ(at)] for all additive endomorphisms ϕ, and all a1, …, at ∈ R, then R possesses a bounded ideal A such that R/A satisfies the polynomial identity f.

Rings whose additive endomorphisms are multiplicative

Rings whose additive endomorphisms are multiplicative

Rings all of whose additive group endomorphisms are left multiplications

Motivated by Cauchy′s functional equation f(x + y) = f(x) + f(y), we study in §1 special rings, namely, rings for which every endomorphism f of their additive group is of the form f(x) ≡ ax. In §2 we generalize to R algebras (R a fixed commutative ring) and give a classification theorem when R is a complete discrete valuation ring. This result has an interesting consequence, Proposition 12, for the theory of special rings.

Additive endomorphisms of rings

PROOF. Assume S----Z2, $ A~ ~ A1 is a member of R, and let ~ be an additive endomorphism of R. We will prove y must be a ring endomorphism of $. The map y will send A1 to A1 and will send Z~ ~ A2 to Z2, @ A 2 since mx = 0 implies that my(x) ~ O. It will then suffice to show that y restricted to Z~ ~B A~ is a ring endomorphism. Therefore we will assume A 1 = 0. Products in S are defined by xy = 2n-ix A 2n-ly where the symbol a A b is defined to be a if a = b and 0 ff a ---- 0 or b = 0. It is undefined otherwise. From the definition of a A b ~(a A b) = ~(a) A ~(b) whenever a A b is defined. Therefore

Unital rings whose additive endomorphisms commute

Unital rings whose additive endomorphisms commute

Groupoids with Additive Endomorphisms

Groupoids with Additive Endomorphisms

Groupoids with Additive Endomorphisms

Groupoids with Additive Endomorphisms