An algebraic structure or algebraic system consists of

• a nonempty set $A$ (called the underlying set, carrier set, or domain),
• a collection of operations on $A$ (typically binary operations, e.g. addition, multiplication),
• a finite set of identities (known as axioms) that the operations must satisfy

To be an algebraic structure, the set must be closed under the operation(s).

The study of algebraic structures is called abstract algebra.
The general theory of algebraic structures is formalized in universal algebra.
Even more generally, category theory includes other mathematical structures.

Common axioms

commutativity : an operation $\ast$ is commutative if $x\ast y=y\ast x$ for every $x$ and $y$
associativity : an operation $\ast$ is associative if $(x\ast y)\ast z=x\ast (y\ast z)$ for every $x$ and $y$

left distributivity : an operation $\ast$ is left-distributive with respect to another operation $+$ if $x\ast (y+z)=(x\ast y)+(x\ast z)$ for every $x,y,z$
right distributivity : an operation $\ast$ is right-distributive with respect to another operation $+$ if $(y+z)\ast x=(y\ast x)+(z\ast x)$ for every $x,y,z$

distributivity : an operation $\ast$ is distributive with respect to another operation $+$ if it is both left-distributive and right-distributive. If $\ast$ is commutative, proving one is sufficient.

identity element : a binary operation $\ast$ has an identity element if there is an element $e$ such that $x\ast e=x$ and $e\ast x=x$ for all $x$

inverse element : for a binary operation $\ast$ with inverse element $e$, an element $x$ is invertible if there is an inverse element $x^{-1}$ such that $x^{-1}\ast x=e$ and $x\ast x^{-1}=e$.

Common algebraic structures

set : a degenerate algebraic structure $S$ having no operations
magma : a set with a binary operation under which it is closed
semigroup : an associative monoid
monoid : a semigroup with an identity element
group : a monoid with an inverse defined for all elements
abelian group : a commutative group
ring : for two binary operations $\ast$ and $+$, the addition is an abelian group, the multiplication is a monoid, and the multiplication is distributive with respect to addition
field
vector space