=====Rings=====

Abbreviation: **Rng**
====Definition====
A \emph{ring} is a structure $\mathbf{R}=\langle R,+,-,0,\cdot
\rangle $ of type $\langle 2,1,0,2\rangle $ such that


$\langle R,+,-,0\rangle $ is an [[abelian groups]]


$\langle R,\cdot \rangle $ is a [[semigroups]]


$\cdot $ distributes over $+$:  $x\cdot (y+z)=x\cdot y+x\cdot z$, $(y+z)\cdot x=y\cdot x+z\cdot x$

==Morphisms==
Let $\mathbf{R}$ and $\mathbf{S}$ be rings. A morphism from $\mathbf{R}$
to $\mathbf{S}$ is a function $h:R\to S$ that is a homomorphism: 

$h(x+y)=h(x)+h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$

Remark: 
It follows that $h(0)=0$ and $h(-x)=-h(x)$.

====Examples====
Example 1: $\langle\mathbb{Z},+,-,0,\cdot\rangle$, the ring of integers with addition, subtraction, zero, and multiplication.


====Basic results====
$0$ is a zero for $\cdot$: $0\cdot x=0$ and $x\cdot 0=0$.

====Properties====
^[[Classtype]]  |variety |
^[[Equational theory]]  |decidable |
^[[Quasiequational theory]]  | |
^[[First-order theory]]  |undecidable |
^[[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====

$\begin{array}{lr}
f(1)= &1\\
f(2)= &2\\
f(3)= &2\\
f(4)= &11\\
f(5)= &2\\
f(6)= &4\\
\end{array}$

[[Finite rings in the Encyclopedia of Integer Sequences]]


====Subclasses====
[[Commutative rings]] 

[[Rings with identity]] 

====Superclasses====
[[Semirings]] 


====References====

[(Ln19xx>
)]