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regular_rings [2010/07/29 15:46] (current)
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 +=====Regular rings=====
 +Abbreviation: **RRng**
 +====Definition====
 +A \emph{regular ring} is a [[rings with identity]] $\mathbf{R}=\langle R,+,-,0,\cdot,1
 +\rangle $ such that
 +
 +every element has a pseudo-inverse:  $\forall x\exists y(x\cdot y\cdot x=x)$
 +
 +==Morphisms==
 +Let $\mathbf{R}$ and $\mathbf{S}$ be regular rings. A morphism from $\mathbf{R}$
 +to $\mathbf{S}$ is a function $h:R\rightarrow S$ that is a homomorphism:
 +
 +$h(x+y)=h(x)+h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(1)=1$
 +
 +Remark:
 +It follows that $h(0)=0$ and $h(-x)=-h(x)$.
 +
 +\begin{examples}
 +\end{examples}
 +====Properties====
 +^[[Classtype]]  |first-order |
 +^[[Equational theory]]  | |
 +^[[Quasiequational theory]]  | |
 +^[[First-order theory]]  | |
 +^[[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====
 +
 +$\begin{array}{lr}
 +f(1)= &1\\
 +f(2)= &\\
 +f(3)= &\\
 +f(4)= &\\
 +f(5)= &\\
 +f(6)= &\\
 +\end{array}$
 +
 +====Subclasses====
 +[[Division rings]]
 +
 +====Superclasses====
 +[[Rings with identity]]
 +
 +
 +====References====
 +
 +[(Ln19xx>
 +)]