A Note on Additive Groups of Some Specific Torsion-Free Rings of Rank Three and Mixed Associative Rings
Journal Title: Discussiones Mathematicae - General Algebra and Applications - Year 2017, Vol 37, Issue 2
Abstract
It is studied how rank two pure subgroups of a torsion-free Abelian group of rank three influences its structure and type set. In particular, the criterion for such a subgroup B to be a direct summand of a torsion-free Abelian group of rank three with the finite type set containing the greatest element which does not belong to the type set of B, is presented. Some results for nil groups and the square subgroup of a decomposable torsion-free Abelian group are also achieved. Moreover, new results for mixed Abelian groups supporting only associative rings are obtained. In particular, the first example of an Abelian group supporting only associative rings but not only commutative rings is given.
Authors and Affiliations
Alireza Najafizadeh, Mateusz Woronowicz
Keywords
Related Articles
- EP ID EP395681
- DOI 10.7151/dmgaa.1274
- Views 23
- Downloads 0
Characterizations of ordered Γ-Abel-Grassmann's groupoids
In this paper, we introduced the concept of ordered Γ-AG-groupoids, Γ- ideals and some classes in ordered Γ-AG-groupoids. We have shown that every Γ-ideal in an ordered Γ-AG∗∗-groupoid S is Γ-prime if and only if it is Γ...
Intervals of certain classes of Z-matrices
Let A and B be M-matrices satisfying A ≤ B and J = [A, B] be the set of all matrices C such that A ≤ C ≤ B, where the order is component wise. It is rather well known that if A is an M-matrix and B is an invertible Mmatr...
LEFT ZEROID AND RIGHT ZEROID ELEMENTS OF Γ-SEMIRINGS
In this paper we introduce the notion of a left zeroid and a right zeroid of Γ-semirings. We prove that, a left zeroid of a simple Γ-semiring M is regular if and only if M is a regular Γ-semiring.
Generalized derivations in prime rings and Banach algebras
Let R be a prime ring with extended centroid C, F a generalized derivation of R and n ≥ 1, m ≥ 1 fixed integers. In this paper we study the situations: 1. (F(x ◦ y))m = (x ◦ y) n for all x, y ∈ I, where I is a nonzero id...
ZERO-DIVISOR GRAPHS OF REDUCED RICKART ∗-RINGS
For a ring A with an involution ∗, the zero-divisor graph of A, Γ∗ (A), is the graph whose vertices are the nonzero left zero-divisors in A such that distinct vertices x and y are adjacent if and only if xy∗ = 0. In this...