commutative_rings - MathStructures

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=====Commutative rings=====

Abbreviation: **CRng**
====Definition====
A \emph{commutative ring} is a [[rings]] $\mathbf{R}=\langle R,+,-,0,\cdot\rangle$ such that

$\cdot$ is commutative:  $x\cdot y=y \cdot x$


Remark:  $Idl(R)=\{ all ideals of R\}$

$I$ is an ideal if $a,b\in I\Longrightarrow a+b\in I$
 
and $\forall r \in R\ (r\cdot I\subseteq I)$


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

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

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 $\cdot$: $0\cdot x=x$ 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====

$\begin{array}{lr}
f(1)= &1\\
f(2)= &2\\
f(3)= &2\\
f(4)= &9\\
f(5)= &2\\
f(6)= &4\\
[http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A037289 Finite commutative rings in the Encyclopedia of Integer Sequences]
\end{array}$

====Subclasses====
[[Commutative rings with identity]] 

[[Fields]] 

====Superclasses====
[[Rings]] 


====References====

[(Ln19xx>
)]

Trace: • representable_lattice-ordered_groups • finite_ordered_commutative_semigroups • cancellative_semigroups • quasigroups • boolean_semigroups • idempotent_semirings_with_identity_and_zero • transitive_relations • bands