Appendix:Glossary of abstract algebra

This is a glossary of abstract algebra

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 $*$, such that, for any operands $a,b,c,(a*b)*c=a*(b*c)$.

C

commutative

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

D

distributive

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

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.

See also

• Appendix:English arities and adicities
• Appendix:English polynomial degrees
• Appendix:Glossary of group theory