Table of Contents
Rings with identity
Abbreviation: Rng$_1$
Definition
A \emph{ring with identity} is a structure $\mathbf{R}=\langle R,+,-,0,\cdot,1 \rangle $ of type $\langle 2,1,0,2,0\rangle $ such that
$\langle R,+,-,0,\cdot\rangle $ is a ring
$1$ is an identity for $\cdot$: $x\cdot 1=x$, $1\cdot x=x$
Morphisms
Let $\mathbf{R}$ and $\mathbf{S}$ be rings with identity. A morphism from $\mathbf{R}$ to $\mathbf{S}$ is a function $h:R\rightarrow S$ that is a homomorphism:
$h(x+y)=h(x)+h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(1)=1$
Remark: It follows that $h(0)=0$ and $h(-x)=-h(x)$.
Examples
Example 1: $\langle\mathbb{Z},+,-,0,\cdot,1\rangle$, the ring of integers with addition, subtraction, zero, multiplication, and one.
Basic results
$0$ is a zero for $\cdot$: $0\cdot x=0$ and $x\cdot 0=0$.
Properties
Finite members
$\begin{array}{lr}
f(1)= &1
f(2)= &1
f(3)= &1
f(4)= &4
f(5)= &1
f(6)= &1
\end{array}$
Finite rings with identity in the Encyclopedia of Integer Sequences
