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

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.