Abelian dqt-Groups and Rings on Them

Abstract

The absolute radical of an Abelian group G is the intersection of radicals of all associative rings with additive group G. The problem of describing absolute radicals was formulated by L. Fuchs. He described the absolute Jacobson radical of a torsion Abelian group. In this work, the absolute Jacobson radical and the absolute nil-radical are investigated in some mixed Abelian group classes.

A Note on Additive Groups of Some Specific Associative Rings

“…First of all, we note that there are different articles which have been written not only by S. Feigelstock and A. Chekhlov but also by other algebraists lately: Pham Thi Thu Thuy, E. Kompantseva, (cf. [12,13,15,16]). Symbols Q, Z, P, N stand for the field of rationals, the ring of integers, the set of all prime numbers, the set of all natural numbers understood as the set of all positive integers, respectively.…”

Multiplication Groups of Abelian Torsion-Free Groups of Finite Rank

“…Since the prime integer m does not divide s 2 1 − s 2 2 , we have that m divides β; therefore, m divides α by (14). This contradicts to the property that (u 1 , u 2 ) / ∈ M (2) .…”

“…For an Abelian group G, a multiplication on G is a homomorphism µ : G⊗G → G. The set Mult G of all multiplications on the group G itself is an Abelian group with respect to addition; the group is called the multiplication group of G or the group of multiplications on G [9]. An Abelian group G with multiplication on G is called a ring on the group G. The problem of studying the relationship between the structure of an Abelian group and the properties of ring structures on it is very multifaceted and has a long history in algebra; see [1], [2], [6], [7], [10], [11], [13], [14].…”

Умножения На Смешанных Абелевых Группах

“…Во втором случае = * Λ . В этой ситуации в силу леммы 1 и того, что максимальная Λ( )-делимая подгруппа группы является абсолютным нильпотентным идеалом [16], для любого кольца ( , ×) на факторкольцо ( / ( ), ×) является ниль-кольцом. Более того, для любого ∈ существует такое ∈ N, что в каждом кольце на степень при любой расстановке скобок принадлежит ( ).…”