# Appendix:Glossary of abstract algebra

Definition from Wiktionary, the free dictionary

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[edit]

- associative
- Of an operator *, such that, for any operands a,b,c, (a * b) * c = a * (b * c).

## C[edit]

- commutative
- Of an operator *, such that, for any operands a,b, a * b = b * a.

## D[edit]

- distributive
- Of an operation * with respect to the operation o, such that a * (b o c) = (a * b) o (a * c).

## F[edit]

- 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[edit]

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

## I[edit]

- 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[edit]

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

## R[edit]

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

## S[edit]

- 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.