regular_rings - MathStructures

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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>
)]

Trace: • lattice-ordered_rings • finite_binars_with_identity • finite_ordered_semilattices • t0-spaces • finite_commutative_binars_with_identity • bilattices • steiner_quasigroups • complete_distributive_lattices • schroeder_categories • bounded_lattices