# Appendix:Glossary of abstract algebra

This is a glossary of abstract algebra

Table of Contents: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

## A

associative
Of an operator ${\displaystyle *}$, such that, for any operands ${\displaystyle a,b,c,(a*b)*c=a*(b*c)}$.

## C

commutative
Of an operator ${\displaystyle *}$*, such that, for any operands ${\displaystyle a,b,a*b=b*a}$.

## D

distributive
Of an operation ${\displaystyle *}$ with respect to the operation ${\displaystyle o}$, such that ${\displaystyle a*(boc)=(a*b)o(a*c)}$.

## F

field
A set having two operations called addition and multiplication under both of which all the elements of the set are commutative and associative; for which multiplication distributes over addition; and for both of which there exist an identity element and an inverse element.

## G

group
A set with an associative binary operation, under which there exists an identity element, and such that each element has an inverse.

## I

ideal
A subring closed under multiplication by its containing ring.
identity element
A member of a structure which, when applied to any other element via a binary operation induces an identity mapping.

## M

monoid
A set which is closed under an associative binary operation, and which contains an element which is an identity for the operation.

## R

ring
An algebraic structure which is a group under addition and a monoid under multiplication.

## S

semigroup
Any set for which there is a binary operation that is both closed and associative.
semiring
An algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse.