A ring is a structure $(R,+,\cdot)$ where
- $(R,+)$ is an abelian group;
- $(R,\cdot)$ is a monoid;
- $\cdot$ is distributive over $+$.
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 $1$.
A ring $(R,+,\cdot)$ is abelian if $(R,\cdot)$ is abelian.
The only ring where $0=1$ is $\{0\}$, the trivial / zero ring.
If $R$ is a non-trivial ring, we usuallly write
\[R^* = R\setminus\{0\}.\]
A unit of a ring is a multiplicatively invertible element.
Let $R$ be a ring, $a\in R$ and $n\in\N$. We define $a^n$ recursively:
- \[a^0 = 1.\]
- \[a^{n+1} = a^n\cdot a.\]
For $n\in\N^+$, we have
\[a^n = \underbrace{a\cdot\cdots\cdot a}_{\text{$n$ times}}.\]
Let $(R,+,\cdot,0,1)$ be a ring. For any $a\in R$,
\[a\cdot 0 = 0\cdot a = 0.\]
Proof.For any $a\in R$,
\[a\cdot 0 = a\cdot(1-1) = a\cdot 1-a\cdot 1 = 0\]and
\[0\cdot a = (1-1)\cdot a = 1\cdot a-1\cdot a = 0.\]
In a non-zero ring, $0$ is not invertible.