P-Adic Localization in Rational Numbers: A Proof of Local Subring Property

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In summary, the conversation discusses the definition of a prime number and the set $R$ consisting of rational numbers with a prime not dividing the denominator. The individual wants to prove that $R$ is a local subring of $\mathbb{Q}$ and wonders if showing the existence of a specific ideal $I$ would suffice. They suggest using the ideal $pR$ and condition 4 to prove the locality of $R$.
  • #1
mathmari
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Hey! :eek:

Let $p$ be a prime.
We define $$R=\{m/n\in \mathbb{Q}\mid m,n\in \mathbb{Z} \text{ and } p\not\mid n\}$$

I want to show that $R$ is a local subring of $\mathbb{Q}$. To show that, do we have to show that there is a $I\subseteq R$ which satisfies the following conditions?
  1. $I$ is the only maximal right ideal of $R$
  2. $I$ is the only maximal left ideal of $R$
  3. $I$ is an ideal
  4. each element $a\in R-I$ is invertible
(Wondering)
 
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  • #2
Consider $I = pR$ and use condition 4 to prove $R$ is local.
 

Related to P-Adic Localization in Rational Numbers: A Proof of Local Subring Property

1. What is a local subring?

A local subring is a subset of a larger ring that satisfies the properties of a subring, but also has the additional property of being a local ring. This means that it has a unique maximal ideal, which is a special type of ideal that is contained in all other ideals of the ring.

2. How is a local subring different from a subring?

A local subring is a type of subring that has the additional property of being a local ring. This means that it has a unique maximal ideal, while a general subring does not necessarily have this property. However, all local subrings are also subrings.

3. What are the uses of local subrings in mathematics?

Local subrings are commonly used in algebraic geometry and commutative algebra, where they play an important role in the study of rings and their ideals. They also have applications in other areas of mathematics, such as number theory and representation theory.

4. How do you determine if a subring is local?

To determine if a subring is local, you need to check if it has a unique maximal ideal. This can be done by examining the ideals of the subring and determining if there is a single ideal that is contained in all others. If so, then the subring is local.

5. Can a local subring have more than one maximal ideal?

No, a local subring can only have one maximal ideal. This is a defining property of local rings and is what sets them apart from other types of rings. If a subring has more than one maximal ideal, it is not considered a local subring.

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