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—		commutative_regular_rings [2010/07/29 15:46] (current)
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		+	=====Commutative regular rings=====
		+	Abbreviation: **CRRng**
		+	====Definition====
		+	A \emph{commutative regular ring} is a [[regular rings]] $\mathbf{R}=\langle R,+,-,0,\cdot,1
		+	\rangle$ such that
		+	$\cdot$ is commutative:  $x\cdot y=y\cdot x$
		+
		+	==Morphisms==
		+	Let $\mathbf{R}$ and $\mathbf{S}$ be commutative regular rings. A morphism from $\mathbf{R}$
		+	to $\mathbf{S}$ is a function $h:R\rightarrow S$ that is a homomorphism:
		+
		+	$h(x+y)=h(x)+h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(1)=1$
		+	====Examples====
		+	Example 1:
		+
		+	====Basic results====
		+
		+	====Properties====
		+	^[[Classtype]]  |first-order |
		+	^[[Equational theory]]  | |
		+	^[[Quasiequational theory]]  | |
		+	^[[First-order theory]]  | |
		+	^[[Locally finite]]  |no |
		+	^[[Residual size]]  |unbounded |
		+	^[[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====
		+
		+	$\begin{array}{lr}
		+	f(1)= &1\\
		+	f(2)= &\\
		+	f(3)= &\\
		+	f(4)= &\\
		+	f(5)= &\\
		+	f(6)= &\\
		+	\end{array}$
		+
		+	====Subclasses====
		+	[[Fields]]
		+
		+	====Superclasses====
		+	[[Commutative rings with identity]]
		+
		+
		+	====References====
		+
		+	[(Ln19xx>
		+	)]