Abstract algebra studies algebraic structures like groups, ring, fields and algebras.

## Group

A group $(X,\cdot)$ is a set with a binary operation that satisfies:

1. Closure: $\forall a,b \in X, a\cdot b \in X$
2. Associativity: $\forall a,b,c \in X, (a \cdot b) \cdot c = a \cdot (b \cdot c)$
3. Identity element: $\exists e \in X, \forall a \in X, e \cdot a = a \cdot e = a$
4. Inverse element: $\forall a \in X, \exists b \in X, b \cdot a = a \cdot b = e$

Basic properties:

• The additive identity and the additive inverse of each element are unique.

Examples of groups:

• Integers $(\mathbb{Z},+)$, where the identity element is 0 and the inverse element of $a$ is $-a$.
• Dihedral groups have underlying sets consisting of symmetries like rotations and flections, and composition as group operation.

### Related concepts

An algebraic structure satisfying:

1. condition 1 is a groupoid.
2. conditions 1-2 is a semigroup.
3. conditions 1-3 is a monoid.

Group operation need not be commutative. A group with commutative operation is call an abelian group.

## Ring

A ring $(X,+,\cdot)$ is a set with two binary operations called addition and multiplication, such that:

1. $(X,+)$ is an abelian group with identity element (denoted by) 0.
2. $(X,\cdot)$ is a monoid with identity element (denoted by) 1.
3. Multiplication left and right distributes over addition:
1. $a \cdot (b + c) = (a \cdot b) + (a \cdot c)$
2. $(a + b) \cdot c = (a \cdot c) + (b \cdot c)$

Basic properties:

• Multiplication by 0 annihilates $X$: $0 \cdot a = a \cdot 0 = 0$
• $-1 \cdot a = -a$
• If $0 = 1$ in a ring, then the ring has only one element.

Examples of rings:

• Integers $(\mathbb{Z},+,\cdot)$, where the additive identity element is 0, the additive inverse element of $a$ is $-a$, and the multiplicative identity element is 1.
• Modular arithmetic $\mathbb{Z}/n\mathbb{Z}$
• $\mathcal{M}_n(R)$, where the underlying set is all n-by-n matrices over an arbitrary ring $R$, with matrix addition and matrix multiplication as corresponding operations. It is a special case of matrix ring.
• Polynomial ring over $R$, $R[t]$, consists of the set of polynomials in one or more variables $t$ with coefficients in another ring $R$, often a field.

### Commutative ring

Ring multiplication is not required to be commutative. Rings that also satisfy commutativity for multiplication are called commutative rings.

### Semiring

A semiring is a similar algebraic structure to a ring, without additive inverse elements.

## Field

A field $(X,+,\cdot)$ is a set with two binary operations called addition and multiplication, such that:

1. $(X,+)$ is an abelian group with identity element (denoted by) 0.
2. $(X \smallsetminus {0},\cdot)$ is an abelian group with identity element (denoted by) 1.
3. Multiplication left and right distributes over addition.

Note:

• A field is equivalent to a commutative ring, with multiplicative inverses for every element except the additive identity element 0.
• Subtraction and division are defined implicitly in terms of the inverse operations of addition and multiplication.

Examples of fields:

• Rational numbers $(\mathbb{Q},+,\cdot)$
• Finite field with four elements.