Appendix:Glossary of abstract algebra

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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.

See also[edit]