All Questions
Tagged with lattice-orders ideals
25
questions
2
votes
1
answer
97
views
The equivalences between points in a locale in constructive mathematics
I am currently following the definitions of the book Frames and Locales and those in the articles Pointfree topology and constructive mathematics and Topo-logie. There are at least three ways of ...
- 73
3
votes
1
answer
44
views
Stone's criterion for distributive lattices
A lattice $L$ is distributive iff for every $a\ne b$ in $L$ there is a prime ideal $P\in\textrm{Spec}(L)$ s.t. either $a\in P\not\ni b$ or $b\in P\not\ni a.$
Right to left. Since $D:L\to2^{\textrm{...
- 1,313
1
vote
1
answer
167
views
Is every modular lattice the ideals of a commutative ring?
Fix a (unital) commutative ring $R$ and let $L(R)$ denote the lattice of ideals of $R$, partially ordered by inclusion. In this answer, rschwieb notes that "the problem of representing lattices ...
- 9,995
3
votes
1
answer
171
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What kind of algebra is the lattice of ideals of a ring?
I have been messing around with rings and ideals for the past week or so in an attempt to prove Nakayama's lemma as an exercise. I completely failed to prove the lemma, but I did notice something ...
- 11.9k
0
votes
0
answers
46
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Canonical form of module in a real quadratic field
Consider a real quadratic field K. Let M be a complete Z-module in K.
I would like to see a proof that it can be multiplied by a totally positive number $x$ so that
$$xM=\mathbb{Z} + w\mathbb{Z}$$
...
- 1,509
4
votes
2
answers
122
views
Is there an intrinsic lattice-theoretic generalization of the set of principal ideals of a ring?
Let $R, S$ be unital commutative rings.
Upon revisiting basics of ring theory, I've been wondering whether there is an intrinsic lattice-theoretic description of what it means to be a principal ideal.
...
- 3,375
5
votes
0
answers
119
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What is the distributive reflection of the lattice of ideals of a ring?
Let $A$ be a commutative ring and let $L$ be the set of ideals of $A$, partially ordered by inclusion. $L$ is a complete lattice but is not distributive in general. For instance, in the polynomial ...
- 90.9k
10
votes
1
answer
279
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Constructing rings with a specific lattice of ideals.
Let $R$ be a commutative ring with 1. The ideals of $R$ form a lattice with inclusion as order relation. Let me call it the ideal lattice $L(R)$ of $R$.
Given an arbitrary lattice $L$, there are some ...
- 30.1k
1
vote
1
answer
33
views
Restricting a filter in a Boolean algebra to a generating set and have it generate a filter
Let $B$ be a Boolean algebra and $S \subseteq B$ be a subset that generates $B$. Is it the case that every filter $x$ of $B$ is equal to the filter generated by $x \cap S$? What if $S$ itself is a ...
- 7,290
2
votes
0
answers
92
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On a necessary condition for being a prime ideal
All rings below are commutative with unity.
If $P$ is a prime ideal in a ring $R$, then it has the following property:
(*) For every ideal $I,J$ of $R$, $I \cap J \subseteq P \implies I \subseteq P$...
user640299
2
votes
1
answer
137
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Characterization of zero-dimensional frames via lattices of ideals
My question concerns the left-to-right implication of the following:
Theorem A frame $L$ is compact and zero-dimensional iff it is isomorphic to the lattice of ideals $\mathcal{I}(B)$ of some Boolean ...
4
votes
2
answers
220
views
The lattice of annihilator ideals of a ring
The question is about an exercise from the book "Lattice-ordered rings and modules" from Stuart A. Steingberg. This is the exercise 7 from chapter 1, section 2.
Let $R$ a ring with no nonzero ...
- 1,017
0
votes
1
answer
38
views
Ideals of partially defined lattices
If $L$ is a (fully defined) lattice, then given a subset $S \subseteq L$, it is known that the ideal $I(S)$ of $L$ generated by the set $S$ has the description $$ I(S) = \{x \in L : x \leq s_1 \vee \...
- 1,621
10
votes
0
answers
519
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Relative chinese remainder theorem and the lattice of ideals
Let $R$ be a ring with two-sided ideals $I,J$. The proof of the "absolute" chinese remainder theorem revolves around the fact that if $I,J$ cover $R$ in the lattice of ideals, i.e $I+J=R$, then the ...
- 13.9k
2
votes
3
answers
220
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A question on Join homomorphism and Ideals
On page 287 of the book Mathematical Methods in Linguistcs, by Barbara Partee, Alice Ter Meulen and Robert E. Wall (Dordrecht, Kluwer Academic Press, 1993), I find the following theorem, which they ...
- 2,053