All Questions
Tagged with lattice-orders ring-theory
27
questions
2
votes
1
answer
45
views
Stone-Cech compactification via lattice ideals of $Coz(X)$
While studying P. T. Johnstone's book Stone Spaces I have come across the following Proposition (Section: 3.3)
Where $max C(X)$ is the set of all maximal ideals of $C(X)$ (ring of real-valued ...
- 1,068
2
votes
1
answer
83
views
The Boolean lattice of a Boolean ring
I am proving that a Boolean Ring is also a Boolean Lattice.
I defined $\leq$ as $x\leq y$ when $xy=x$. The supremum is $a+b+ab$, and infimum is $ab$. The Max element is $1$, Min is $0$.
I proved that $...
- 319
3
votes
1
answer
171
views
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
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
2
votes
0
answers
92
views
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
100
views
Proving associativity of a certain binary aperation in any complemented distributive lattice
If, in a Boolean lattice $(X,\vee,\wedge,0,1,')$ (i.e. a complemented distributive lattice), we define $x+y=(x'\wedge y)\vee(x\wedge y')$, is there an elegant way to prove that $(x+y)+z=x+(y+z)$ ...
- 1,017
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
1
vote
1
answer
80
views
Distributive lattices
I have a question which is in my ring theory lesson. it's under the topic of distributive lattice and I don't know how to prove it.
Que: If A is a strongly regular ring, then the principle right ...
- 11
2
votes
1
answer
193
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What algebraic structure best fits the Cantor set?
Let $2^{\Bbb N}$ be the set of infinite binary sequences $\{x_n\}$ where $x_n\in \{0,1\}$ for every $n\in \Bbb N$. I want it to fit the axioms of a known algebraic structure such that the following ...
- 3,226
2
votes
0
answers
37
views
Complete positively ordered rigs (semirings)
A positively ordered rig $A$ is a rig together with a (pre-)order $\leq$ such that:
$x\leq y \Rightarrow x + z \leq y +z$
$x\leq y \Rightarrow x\cdot z \leq y\cdot z\,\&\, z\cdot x \leq z\cdot y$
...
- 12.5k
1
vote
2
answers
146
views
Cardinality of the base of a ring of sets
Concretely, my question is:
What is the size of the minimal base of a ring of sets? In my understanding the base is the set of elements that can be used to "deduce the rest". E.g., if I have a ring ...
- 11
1
vote
1
answer
179
views
Boolean lattices vs boolean rings
Which kinds of theorems about boolean algebras are easier to prove with boolean rings (than with actual boolean lattices)?
Give me at least one example, as an answer.
- 5,103
1
vote
1
answer
34
views
How to find an $R$-order $\Lambda$, such that $A=k\Lambda$?
Let $R$ be a Noetherian integral domain with quotient field $k$.
An $R$-order $\Lambda$ is a finitely generated torsionless non-zero $R$-module, which is at the same time an $R$-algebra.
Why is ...
- 540
2
votes
1
answer
254
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Subrings of the product of rings of algebraic integers
Let $K,L$ be a number fields and $\mathcal{O}_K,\mathcal{O}_L$ their rings of algebraic integers, and $n_K=[K:\mathbb Q]$, $n_L=[L:\mathbb Q]$.
Suppose that $R$ is a subring of $\mathcal{O}_K\oplus\...
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