Appendix:Glossary of abstract algebra

This is a glossary of abstract algebra

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 $$a * (b o c) = (a * b) o (a * c)$$.

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.