Appendix:Glossary of abstract algebra

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Glossary of abstract algebra

Wikipedia

This is a glossary of abstract algebra

associative

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

commutative

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

distributive

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

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.

group

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

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.

monoid

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

ring

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

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.

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