Ring Theory

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Rings (and modules) are defined by decorating Abelian Groups with additional data. As motivation for the introduction of such structures, note that all number-based examples of groups that we have encountered, such as $\mathbb{Z}$ or $\mathbb{R}$, are endowed with an operation of multiplication as well as the ‘addition’ making them into (abelian) groups.

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Notes

Basic Definitions

• Ring
  • Examples of Rings
  • Ring Homomorphisms
• Commutative Ring
• Integral Domain
  • Zero Divisors
• Division Ring
• Fields
  • Units (Rings)
  • Finite Integral Domains are Fields
• Polynomial Rings
• Monoid Ring

The Category of Rings

• The Category Ring
• Universal Property of Polynomial Rings
• Monomorphisms and Epimorphisms in Rings
• Products in Rings

Ideals and Quotient Rings

• Ideals
• Quotient Rings
• Cannonical Decomposition of Ring Homomorphisms
• Ideals in Quotients
• Noetherian Ring and PIDs
  • Principle Ideals
• Quotients of Polynomial Rings
  • Quotienting by monic polynomial is an abelian group isomorphism with direct sum
  • Complex Numbers as quotients of polynomial ring over Reals
• Prime Ideals and Maximal Ideals
  • Prime Ideals are Maximal in PIDs
• Spectrum of a Ring
  • Spectrums in PIDs
  • Krull Dimension

Existence of Factors

• Ascending Chain Condition
• Hilbert’s Basis Theorem
• Polynomial Rings over a Noetherian Ring are Noetherian
• Associates in Rings
• Prime and Irreducible elements
• Prime elements are irreducible

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MOCs

• Category Theory
• Module Theory
• Homology Theory

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References