rings_with_identity - MathStructures

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=====Rings with identity=====

Abbreviation: **Rng**$_1$


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


$\langle R,+,-,0,\cdot\rangle $ is a [[ring]]


$1$ is an identity for $\cdot$:  $x\cdot 1=x$, $1\cdot x=x$

==Morphisms==
Let $\mathbf{R}$ and $\mathbf{S}$ be 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 $\cdot$: $0\cdot x=0$ 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====

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

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


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

[[Division rings]] 

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


====References====

[(Ln19xx>
)]

Trace: • division_algebras • medial_groupoids • priestley_spaces • rectangular_bands • continuous_posets • complete_lattices • integral_domains • euclidean_domains • hausdorff_spaces • finite_groupoids