=====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>
)]