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Unique factorization domain and principal ideals
abstract-algebra
ring-theory
ideals
unique-factorization-domains
Irreducibles are prime in a UFD
abstract-algebra
ring-theory
proof-explanation
unique-factorization-domains
Prove that a UFD is a PID if and only if every nonzero prime ideal is maximal
abstract-algebra
principal-ideal-domains
unique-factorization-domains
Prove that $k[x,y,z,w]/(xy-zw)$, the coordinate ring of $V(xy-zw) \subset \mathbb{A}^4$, is not a unique factorization domain
abstract-algebra
algebraic-geometry
commutative-algebra
unique-factorization-domains
Show that $k[x,y,z]/(xz-y^2)$ is not a UFD.
ring-theory
unique-factorization-domains
Where Fermat's last theorem fails
abstract-algebra
number-theory
elementary-number-theory
integral-domain
unique-factorization-domains
Principal prime ideals are minimal among prime ideals in a UFD
commutative-algebra
unique-factorization-domains
Ring of integers is a PID but not a Euclidean domain
abstract-algebra
algebraic-number-theory
principal-ideal-domains
unique-factorization-domains
Motivation for Eisenstein Criterion
abstract-algebra
polynomials
irreducible-polynomials
motivation
unique-factorization-domains
Does $A$ a UFD imply that $A[T]$ is also a UFD?
abstract-algebra
commutative-algebra
unique-factorization-domains
One-dimensional [Noetherian] UFD is a PID
reference-request
commutative-algebra
principal-ideal-domains
unique-factorization-domains
dedekind-domain
About the localization of a UFD
abstract-algebra
commutative-algebra
unique-factorization-domains
Ring of trigonometric functions with real coefficients
abstract-algebra
ring-theory
commutative-algebra
principal-ideal-domains
unique-factorization-domains
Motivation behind the definition of GCD and LCM
abstract-algebra
elementary-number-theory
gcd-and-lcm
unique-factorization-domains
motivation
Why is $\mathbb{Z}[\sqrt{-n}], n\ge 3$ not a UFD?
abstract-algebra
ring-theory
unique-factorization-domains
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