# 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.

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