# Differences

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integral_domains [2010/07/29 15:46] (current)
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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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