# Tagged Questions

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### Monoid Ring of a Commutative Cancellative Ordered Monoid

Suppose $M$ is a commutative cancellative monoid with $0$ as the identity and $+$ as the operation, and $M$ is equipped with an order $\preceq$ defined by  m \preceq m' \text{ if and only if there ...
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### Existence of totally ordered ring with zero divisors

Does there exist a totally ordered ring with zero divisors? I can't think of an example right now.
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### Existence of non-commutative ordered ring

Does there exist a totally ordered ring which is non-commutative? I have searched the web, but I have not found any examples.
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### Does there exist an ordered ring, with $\mathbb{Z}$ as an ordered subring, such that some ring of p-adic integers can be formed as a quotient ring?

I'm not sure if this might instead be MathOverflow material, but I'll give it a shot here first anyway. Does any ring $R$ exist that satisfies the following properties? $R$ is a totally ordered, ...
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### Poset of idempotents

Let $A$ be an associative algebra and consider the poset $I$ of idempotents in $A$, where as usual if $e,f \in A$ we say $e \leq f$ provided that $e = fef$. Clearly $0 \in I$ is minimal, but in ...
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### Why is a commutative ring with an infinite number of idempotent elements unstable?

In a book of model theory I found the following statement: A commutative ring with an infinite number of idempotent elements unstable. I haven't manage to prove it yet. As stability in the model ...
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### Are ideals in rings and lattices related?

There are (at least) two notions of ideals: An ideal in a ring is a set closed under addition and multiplication by arbitrary element. An ideal in a lattice is a set closed under taking smaller ...
### Examples of rings with ideal lattice isomorphic to $M_3$, $N_5$
The notions of a filter and an ideal on a poset make intuitive sense to me, and I can understand why they are dual: A subset $I\subset P$ of a poset $P$ is an ideal if: for all $x\in I$, $y\leq x$ ...