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Chapter 14: Problem 5

Es seien \(R\) ein kommutativer Ring mit \(1, H\) eine Unterhalbgruppe von \((R, \cdot)\) mit \(0 \notin H\) und \(A\) ein Ideal von \(R\) mit \(A \cap H=\emptyset\). Man zeige: (a) Die Menge \(\mathfrak{X}\) aller \(A\) umfassenden, zu \(H\) disjunkten Ideale von \(R\) besitzt bzgl. \(\subseteq\) maximale Elemente. Hinweis: Zorn'sches Lemma. (b) Die maximalen Elemente von \(\mathfrak{X}\) sind Primideale.

Short Answer

Expert verified
Maximal elements exist by Zorn's Lemma, and they are prime ideals.

Step by step solution

01

Understanding the Problem

We are given a commutative ring \( R \) with identity and a subgroup \( H \) of the multiplicative group of elements of \( R \) excluding zero. There's also an ideal \( A \) such that \( A \cap H = \emptyset \). The task is to show that the set \( \mathfrak{X} \) of ideals containing \( A \) and disjoint from \( H \) has maximal elements and these maximal elements are prime ideals.
02

Define the Partially Ordered Set

Let \( \mathfrak{X} \) be the set of all ideals \( J \) of \( R \) such that \( A \subseteq J \) and \( J \cap H = \emptyset \). We order \( \mathfrak{X} \) by inclusion, i.e., \( J_1 \leq J_2 \) iff \( J_1 \subseteq J_2 \).
03

Apply Zorn's Lemma

Zorn's Lemma states that if every chain (i.e., totally ordered subset) in a partially ordered set has an upper bound in the set, then the set contains at least one maximal element. For \( \mathfrak{X} \), take any chain \( \{ J_i \} \), where each \( J_i \) is an ideal. The union \( J = \bigcup J_i \) is an upper bound for this chain in \( \mathfrak{X} \): it contains \( A \) and is disjoint from \( H \) since each \( J_i \cap H = \emptyset \). Thus, by Zorn's Lemma, a maximal ideal in \( \mathfrak{X} \) exists.
04

Show That Maximal Elements are Prime Ideals

Assume \( J \) is a maximal element of \( \mathfrak{X} \). To show \( J \) is prime, consider \( a, b \in R \) such that \( ab \in J \) but \( a otin J \) and \( b otin J \). Then the ideals \( J + (a) \) and \( J + (b) \) must intersect \( H \), as \( J \) is maximal. Let \( h_1 \in J + (a) \cap H \) and \( h_2 \in J + (b) \cap H \). Thus, there exist \( j_1, j_2 \in J \) such that \( h_1 = j_1 + ra \) and \( h_2 = j_2 + sb \). Calculating the product \( h_1h_2 = (j_1 + ra)(j_2 + sb) \) results in an element of \( J \), contradicting \( J \cap H = \emptyset \), unless either \( a \in J \) or \( b \in J \), proving \( J \) is prime.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Commutative Ring
A commutative ring is a fundamental concept in abstract algebra, and it combines both the idea of a ring and the property of commutativity for multiplication. Simply put, a ring is a set equipped with two operations: addition and multiplication, where each operation adheres to specific rules or axioms.

The defining feature of a commutative ring is that the multiplication operation is commutative, meaning that for any two elements \( a \) and \( b \) in the ring, the equation \( ab = ba \) holds true.

Some characteristics of commutative rings include:
  • An additive identity (typically denoted as 0) such that for any element \( a \) in the ring, \( a + 0 = a \).
  • A multiplicative identity (typically denoted as 1) such that for any element \( a \) in the ring, \( a \cdot 1 = a \).
  • The presence of additive inverses for each element, meaning for any element \( a \), there exists \( -a \) such that \( a + (-a) = 0 \).
Commutative rings are useful in various areas of mathematics because they generalize the familiar properties of numbers and make them applicable in more abstract settings.
Prime Ideal
In the context of ring theory, a prime ideal is a special type of ideal with an exceptionally important property. An ideal, in general, is a subset of a ring that absorbs multiplication by ring elements and contains elements' sums. But a prime ideal adds an intriguing twist.

For an ideal \( P \) in a commutative ring \( R \), it is considered prime if:
  • It is not the entire ring \( R \). This means \( P eq R \).
  • Whenever the product of two elements, \( ab \), is in \( P \), at least one of those elements must also be in \( P \). In other words, if \( ab \in P \), then \( a \in P \) or \( b \in P \). This mirrors the property of prime numbers whereby if a prime number divides a product of numbers, it must divide at least one of those numbers.
Prime ideals play a significant role in algebraic geometry and number theory, analogous to prime numbers in arithmetic.
Partially Ordered Set
A partially ordered set (or poset, for short) is a fundamental structure in mathematics that generalizes the familiar idea of order or arrangement. In a poset, not every pair of elements needs to be comparable, which sets it apart from a totally ordered set where each pair is comparable.

For a set \( S \) with a binary relation \( \le \), it is considered a partially ordered set if the following conditions hold:
  • Reflexivity - For every element \( a \) in \( S \), \( a \le a \).
  • Antisymmetry - For any two elements \( a, b \) in \( S \), if \( a \le b \) and \( b \le a \), then \( a = b \).
  • Transitivity - For any three elements \( a, b, c \) in \( S \), if \( a \le b \) and \( b \le c \), then \( a \le c \).
Posets are a powerful tool in various fields of study such as graph theory, computer science, and logic, offering the ability to model hierarchical structures and dependency relationships.
Ideal
In the study of rings, an ideal is a crucial concept that extends the notion of evenness to abstract algebra. An ideal is a subset of a ring that absorbs multiplication from ring elements, allowing the creation of more complex algebraic structures.

The defining property of an ideal \( I \) in a ring \( R \) is that it is closed under addition and absorbs multiplication by elements from \( R \). This means:
  • If \( a, b \) are elements of \( I \), then their sum \( a + b \) is also in \( I \).
  • If \( a \) is an element in \( I \) and \( r \) is an element in \( R \), then their product \( ra \) is also in \( I \).
Ideals are used to define important constructs in ring theory, such as quotient rings, which lead to a deeper understanding of ring properties and their applications.

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Most popular questions from this chapter

Lokalisierung. Es sei \(R\) ein Integrit盲tsbereich mit dem Primideal \(P .\) Zeigen Sie: (a) \(S:=R \backslash P\) ist eine Unterhalbgruppe von \((R, \cdot)\). (b) \(M:=\left\\{\frac{p}{s} \mid p \in P, s \in S\right\\}\) ist die Menge der Nichteinheiten von \(R_{S}\). (c) \(M\) ist ein Ideal von \(R_{S}\), das alle Ideale \(\neq R_{S}\) von \(R_{S}\) umfasst. (d) Im Fall \(R:=\mathbb{Z}, P:=p \mathbb{Z}\) f眉r eine Primzahl \(p\), gilt \(R_{S} / M \cong \mathbb{Z}_{p}\)

Es sei \(R\) ein kommutativer Ring mit dem Ideal \(A .\) Man nennt \(\sqrt{A}:=\left\\{a \in R \mid a^{n} \in A\right.\) f眉r ein \(n \in \mathbb{N}\\}\) das Radikal von \(A\). Man nennt \(A\) reduziert, wenn \(A=\sqrt{A}\) gilt. Zeigen Sie: (a) \(\sqrt{A}\) ist ein Ideal in \(R\). (b) \(A\) ist genau dann reduziert, wenn \(R / A\) keine nilpotenten Elemente \(\neq 0\) besitzt. Dabei hei脽t ein Element \(a\) eines Ringes nilpotent, wenn \(a^{n}=0\) f眉r ein \(n \in \mathbb{N}\). (c) Primideale sind reduziert. (d) Bestimmen Sie das sogenannte Nilradikal \(N:=\sqrt{(0)}\) in \(\mathbb{Z}_{n}\) sowie \(\mathbb{Z}_{n} / N\).

Beweisen Sie die folgende Verallgemeinerung des chinesischen Restsatzes 6.8: Es sei \(R\) ein Ring mit 1, und es seien \(A_{1}, \ldots, A_{n}\) Ideale von \(R\) mit \(A_{i}+A_{j}=R\) f眉r alle \(i \neq j\) (paarweise Teilerfremdheit). Dann ist die Abbildung $$ \psi:\left\\{\begin{array}{ccc} R /\left(A_{1} \cap \cdots \cap A_{n}\right) & \rightarrow & R / A_{1} \times \cdots \times R / A_{n} \\ a+A_{1} \cap \cdots \cap A_{n} & \mapsto & \left(a+A_{1}, \ldots, a+A_{n}\right) \end{array}\right. $$ ein Ringisomorphismus.

Bestimmen Sie f眉r die folgenden Ideale \(A\) von \(R\) ein Element \(a \in R\) mit \(A=(a)\) : (a) \(R=\mathbb{Z}, A=(3,8,9)\). (b) \(R=\mathbb{Z}_{18}, A=(\overline{3}, \overline{8}, \overline{9})\)

Zeigen Sie: Jede Untergruppe der additiven Gruppe \(\mathbb{Z} / n \mathbb{Z}(n \in \mathbb{N})\) ist ein Ideal des Ringes \(\mathbb{Z} / n \mathbb{Z}\). Bestimmen Sie die maximalen Ideale von \(\mathbb{Z} / n \mathbb{Z}\).
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