# Differences

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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> | ||

+ | )] |

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