Ring

Definition 1. A ring is a structure $(R, +, \cdot)$ such that:

1. $(R, +)$ is an abelian group.

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

3. (Distributivity) For all $a$, $x$, $y\in R$,

   \[\begin{align*} a(x + y) &= ax + ay, \\ (x + y)a &= xa + ya. \end{align*}\]

• The neutral element of $(R, +)$, called the additive neutral element, is typically denoted by $0$.

• The neutral element of $(R, \cdot)$, called the multiplicative neutral element, is typically denoted by $1$.

Definition 2. A ring $(R, +, \cdot)$ is abelian if $(R, \cdot)$ is abelian.

Definition 3. A unit of a ring is a multiplicatively invertible element.

Definition 4. Let $R$ be a ring, $a \in R$ and $n \in \N$. $a^n$ is defined recursively:

1. \[a^0 = 1.\]
2. \[a^{n + 1} = a^n \cdot a.\]

For $n \in \N^+$, we have

\[a^n = \underbrace{a \cdot \cdots \cdot a}_{\text{$n$ times}}.\]

Proposition 1. In a ring $R$, $0$ is an absorbing element with respect to multiplication: For all $a \in R$,

\[a \cdot 0 = 0 \cdot a = 0.\]

Proposition 2. In a non-zero ring $R$, $0$ is not invertible.