Ring

A ring (ring with unit) $(R, +, \cdot, 1)$ is a set $R$ with two maps, $+:R \times R \to R$ and $\cdot : R \times R \to R$, called addition and multiplication, and an element $1 \in R$, called the unit, such that the following ring axioms hold:

1. $(R, +)$ is a commutative group where the neutral element is denoted with $0$.

2. $(R, \cdot)$ is a monoid, meaning that:

   1. Associativity: $\forall a, b, c ∈ R: a · (b · c) = (a · b) · c$

   2. Neutral element: $\exists 1 \in R \forall a \in R: 1 \cdot a = a$

3. Distributivity: $\forall a, b, c \in R: a \cdot (b + c) = a \cdot b + a \cdot c$

If the multiplication is commutative the ring is called commutative ring.

If $(R, +, ·, 1)$ is a ring with unit, and $R' ⊂ R$ is a subset of $R$ such that the restriction of addition and multiplication to $R'$ define a ring with addition $+ : R' × R' → R'$, multiplication $· : R' ×R' → R'$ and unit $1$ on $R'$, then $(R', +, ·, 1)$ is called a subring of $(R, +, ·, 1)$.

Examples of rings:

• The set $Z$ of integers with the usual addition and multiplication

• The set $R[x]$of polynomials with coefficients in $R$ with usual polynomial addition and multiplication