## Commutative rings

Abbreviation: CRng

### Definition

A commutative ring is a rings $\mathbf{R}=\langle R,+,-,0,\cdot\rangle$ such that

$\cdot$ is commutative: $x\cdot y=y \cdot x$

Remark: $Idl(R)=\{ all ideals of R\}$

$I$ is an ideal if $a,b\in I\Longrightarrow a+b\in I$

and $\forall r \in R\ (r\cdot I\subseteq I)$

##### Morphisms

Let $\mathbf{R}$ and $\mathbf{S}$ be commutative rings with identity. A morphism from $\mathbf{R}$ to $\mathbf{S}$ is a function $h:R\rightarrow S$ that is a homomorphism:

$h(x+y)=h(x)+h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$

Remark: It follows that $h(0)=0$ and $h(-x)=-h(x)$.

### Examples

Example 1: $\langle\mathbb{Z},+,-,0,\cdot\rangle$, the ring of integers with addition, subtraction, zero, and multiplication.

### Basic results

$0$ is a zero for $\cdot$: $0\cdot x=x$ and $x\cdot 0=0$.

### Properties

Classtype variety decidable undecidable no unbounded no yes yes, $n=2$ yes yes

### Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &2\\ f(3)= &2\\ f(4)= &9\\ f(5)= &2\\ f(6)= &4\\ [http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A037289 Finite commutative rings in the Encyclopedia of Integer Sequences] \end{array}$