# Differences

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commutative_rings [2010/07/29 15:46] (current)
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+=====Commutative rings=====
+Abbreviation: **CRng**
+====Definition====
+A \emph{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 |
+^[[Equational theory]]  |decidable |
+^[[Quasiequational theory]]  | |
+^[[First-order theory]]  |undecidable |
+^[[Locally finite]]  |no |
+^[[Residual size]]  |unbounded |
+^[[Congruence distributive]]  |no |
+^[[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====
+
+$\begin{array}{lr} +f(1)= &1\\ +f(2)= &2\\ +f(3)= &2\\ +f(4)= &9\\ +f(5)= &2\\ +f(6)= &4\\ +[http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A037289 Finite commutative rings in the Encyclopedia of Integer Sequences] +\end{array}$
+
+====Subclasses====
+[[Commutative rings with identity]]
+
+[[Fields]]
+
+====Superclasses====
+[[Rings]]
+
+
+====References====
+
+[(Ln19xx>
+)]

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