Abbreviation: OreDom

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

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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.

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: $h(x \ldots y)=h(x) \ldots h(y)$

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:

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.

$\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}$

[[...]] subvariety

[[...]] expansion

[[...]] supervariety

[[...]] subreduct