List Of Commutative Algebra Topics

• commutative ring
• ring ideal, maximal ideal, prime ideal
• ring monomorphism, ring epimorphism, ring isomorphism
• integral domain
• quotient field
• product of rings
• Boolean algebra
• zero divisor
• nilpotent, reduced (ring theory)
• annihilator
• principal ideal domain
• unique factorization domain
• Dedekind domain
• Fractional ideal
• Ideal class group
• Euclidean domain
• integral closure
• Chinese remainder theorem
• radical of an ideal
• Noetherian ring
• Hilbert's basis theorem
• Hilbert's Nullstellensatz
• Krull dimension
• Lasker-Noether theorem
• formal power series
• local ring
• localization of a ring
• localization of a module
• flat module
• flat map
• projective module
• injective module
• regular local ring
• discrete valuation ring
• regular sequence (algebra)
• Cohen-Macaulay ring
• Gorenstein ring
• complete intersection ring
• Zariski tangent space
• Khler differential
• Dual numbers
• Height (ring theory)
• Krull's principal ideal theorem
• Weierstrass preparation theorem
• Quillen-Suslin theorem
• Koszul complex
• tensor product of fields
• tensor product of R-algebras
• Valuation (mathematics)
• Discrete valuation
• Discrete valuation ring
• I-adic topology
• Spectrum of a ring.

Active research areas

• Serre's multiplicity conjectures
• Homological conjectures

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