commutative_rings_with_identity

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=====Commutative rings with identity=====

Abbreviation: **CRng$_1$**
====Definition====
A \emph{commutative ring with identity} is a [[rings with identity]] $\mathbf{R}=\langle R,+,-,0,\cdot,1
\rangle$ such that 
$\cdot$ is commutative:  $x\cdot y=y\cdot x$

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

$h(x+y)=h(x)+h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(1)=1$

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

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


====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====

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

====Subclasses====
[[Boolean algebras]] 

[[Integral domains]] 

====Superclasses====
[[Commutative rings]] 

[[Rings with identity]] 


====References====

[(Ln19xx>
)]

Trace: • finite_semilattices • topological_groups • bounded_distributive_lattices • commutative_semigroups • commutative_rings_with_identity • notation_and_terminology • weakly_associative_relation_algebras • directed_graphs