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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> | ||
+ | )] |
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