1.3 Principal, prime, and maximal ideals

Ideals are crucial structures in ring theory, defining subsets closed under addition and multiplication by ring elements. They come in various types, including principal, prime, and maximal, each with unique properties and relationships to the ring's structure.

Understanding ideals is key to grasping advanced concepts in algebra. They're used to analyze ring properties, determine polynomial irreducibility, and apply important theorems like the Chinese Remainder Theorem. Mastering ideals opens doors to deeper algebraic insights.

Fundamental Concepts of Ideals

Types of ideals in rings

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• Principal ideals
  • Generated by single ring element $I = (a) = \{ra : r \in R\}$ where $a$ generates
  • Represent multiples of generator (even integers in $\mathbb{Z}$)
• Prime ideals
  • $ab \in P$ implies $a \in P$ or $b \in P$ for any $a, b \in R$
  • Yield integral domain when ring divided by ideal $R/P$
• Maximal ideals
  • Proper ideal not contained in any other proper ideal
  • Produce field when ring divided by ideal $R/M$

Examples of ring ideals

• Principal ideals
  • $\mathbb{Z}$: $(2) = \{2n : n \in \mathbb{Z}\}$ all even integers
  • Polynomial rings: $(x^2 + 1)$ in $\mathbb{R}[x]$ polynomials divisible by $x^2 + 1$
• Prime ideals
  • $\mathbb{Z}$: $(p)$ where $p$ prime number (2, 3, 5, 7, 11)
  • $\mathbb{Z}[x]$: $(x)$ prime but not maximal, contains all polynomials with no constant term
• Maximal ideals
  • $\mathbb{Z}$: $(p)$ where $p$ prime number (same as prime ideals in $\mathbb{Z}$)
  • $\mathbb{R}[x]$: $(x^2 + 1)$ maximal, contains all polynomials divisible by $x^2 + 1$

Relationships between ideal types

• Hierarchy of ideals
  • Maximal ideals always prime, prime not always maximal
  • Principal ideals can be prime, maximal, both, or neither
• Zero ideal $(0)$
  • Prime in integral domains, maximal only in fields
• Relationship to ring properties
  • Principal ideal domains (PIDs) every ideal principal
  • Unique factorization domains (UFDs) generalize PIDs
  • Noetherian rings have ascending chain condition on ideals

Applications of ideal concepts

• Determine ideal type principal, prime, or maximal
• Analyze ideals using quotient ring structure
• Prove ideal properties using definitions
• Study ring structure through prime and maximal ideals
• Apply Chinese Remainder Theorem with maximal ideals
• Investigate prime ideals in polynomial rings
• Determine polynomial irreducibility using maximal ideals