1
vote
0answers
33 views

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 ...
3
votes
1answer
53 views

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.
5
votes
1answer
59 views

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.
8
votes
1answer
115 views

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, ...
6
votes
1answer
135 views

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 ...
5
votes
2answers
248 views

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 ...
8
votes
2answers
381 views

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 ...
11
votes
3answers
581 views

Examples of rings with ideal lattice isomorphic to $M_3$, $N_5$

In thinking about this recent question, I was reading about distributive lattices, and the Wikipedia article includes a very interesting characterization: A lattice is distributive if and only if ...
4
votes
1answer
153 views

Ideals and filters

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 $x\in I$, $y\leq x$ ...