## Ore domains

Abbreviation: OreDom

### Definition

An \emph{Ore domain} is a ring with identity $\mathbf{A}=\langle A,+,-,0,\cdot,1\rangle$ such that

$\cdot$ is \emph{integral}: $xy=0\Longrightarrow x=0\text{ or }y=0$

nonzero common multiples exist: $x\ne 0\ne y\Longrightarrow\exists u\exists v(xu=yv\ne 0)\text{ and }\exists u\exists v(ux=vy\ne 0)$

It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.

##### Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be … . A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(x \ldots y)=h(x) \ldots h(y)$

### Definition

An \emph{…} is a structure $\mathbf{A}=\langle A,\ldots\rangle$ of type $\langle …\rangle$ such that

$\ldots$ is …: $axiom$

$\ldots$ is …: $axiom$

Example 1:

### Properties

Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

Classtype (value, see description) 1)

### Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ \end{array}$ $\begin{array}{lr} f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$

### Subclasses

[[...]] subvariety
[[...]] expansion

### Superclasses

[[...]] supervariety
[[...]] subreduct

1) F. Lastname, \emph{Title}, Journal, \textbf{1}, 23–45 MRreview

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