integral_domains - MathStructures

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=====Integral Domain=====

Abbreviation: **IntDom**
====Definition====
An \emph{integral domain} is a [[commutative rings with identity]] $\mathbf{R}=\langle R,+,-,0,\cdot,1\rangle$ that 


has no zero divisors:  
$\forall x,y\ (x\cdot y=0\Longrightarrow x=0\ \mbox{or}\ y=0)$

==Morphisms==
Let $\mathbf{R}$ and $\mathbf{S}$ be integral domains. 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, and multiplication is an integral domain.


====Basic results====
Every finite integral domain is a [[fields]].

====Properties====
^[[Classtype]]  |Universal class |
^[[Equational theory]]  | |
^[[Quasiequational theory]]  | |
^[[First-order theory]]  | |
^[[Locally finite]]  | |
^[[Residual size]]  | |
^[[Congruence distributive]]  | |
^[[Congruence modular]]  |yes |
^[[Congruence n-permutable]]  |yes, $n=2$ |
^[[Congruence regular]]  |yes |
^[[Congruence uniform]]  |yes |
^[[Congruence extension property]]  | |
^[[Definable principal congruences]]  | |
^[[Equationally def. pr. cong.]]  | |
^[[Amalgamation property]]  | |
^[[Strong amalgamation property]]  | |
^[[Epimorphisms are surjective]]  | |
====Finite members====

$\begin{array}{lr}
f(1)= &1\\
f(2)= &1\\
f(3)= &1\\
f(4)= &1\\
f(5)= &1\\
f(6)= &0\\
\end{array}$

====Subclasses====
[[Unique factorization domains]] 

====Superclasses====
[[Commutative rings with identity]] 


====References====

[(Ln19xx>
)]

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