# Associativity forcing commutativity in left nil rings

#### Abstract

Our study in this thesis is concentrated on the AFC groups (Associativity Forces Commutativity) and observing the ring structure there. We are looking for abelian groups yielding ring structures (K) where associativity forces commutativity. We call them K-AFC groups. We consider that the only conditions put on a ring multiplication are the distributive laws over the additive group addition. After [1], We say that an abelian group (G, +) is an AFC-group if (i) there exists a nonassociative and noncommutative ring (G, +, ˙) and (ii) all associative rings (G, +, ˙) are commutative. We call a ring left-nil if its “left power sequence” contains only a finite number of nonzero terms, We first comment on the structure of one sided-nil rings and then study the left-nil AFC groups which are a type of a K-AFC group. We say that G is a left-nil AFC group if G satisfies the two conditions of K -AFC groups, where K is the class of left-nil rings. The motivation of this study comes from the “Order Algebraic Structures”, where the issue of K-AFC group naturally arose and there is a considerable interest in this topic to investigate a relation between associativity and commutativity.

#### Subject Area

Mathematics

#### Recommended Citation

Bhuiyan, Md Al Masum, "Associativity forcing commutativity in left nil rings" (2015). *ETD Collection for University of Texas, El Paso*. AAI1600304.

https://digitalcommons.utep.edu/dissertations/AAI1600304