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product of two elements that factor in a unique way factors in a unique way?

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#abstract-algebra #ring-theory #unique-factorization-domains
362
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Intuitively, what does being a UFD have to do with line bundles/first homology

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#algebraic-geometry #commutative-algebra #unique-factorization-domains
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When are quadratic rings of integers unique factorization domains?

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#number-theory #commutative-algebra #integers #unique-factorization-domains
80
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Properties of rings of the form $\mathbb{Z}+\mathbb{Z}\sqrt{p}i+\mathbb{Z}\sqrt{q}j+\mathbb{Z}\sqrt{r}k$

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#abstract-algebra #unique-factorization-domains #euclidean-domain
811
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Prove that $5$ is irreducible in $\mathbb Z[\sqrt{2}]$

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#abstract-algebra #ring-theory #unique-factorization-domains
119
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An element lies in a prime ideal but not in its square

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#abstract-algebra #ideals #maximal-and-prime-ideals #unique-factorization-domains
161
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Multiplicative group of the fraction field of a UFD

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#abstract-algebra #group-theory #ring-theory #unique-factorization-domains
172
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Some counterexamples in basic ring theory

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#abstract-algebra #ring-theory #principal-ideal-domains #unique-factorization-domains
280
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Show that $\mathscr{O}_{\mathbb{Q}(\sqrt{-7})}$ is a UFD

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#number-theory #ring-theory #algebraic-number-theory #unique-factorization-domains #euclidean-domain
760
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When is a Unique Factorization Domain a Principal Ideal domain

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#abstract-algebra #ring-theory #principal-ideal-domains #unique-factorization-domains
29
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Trouble understanding proof regarding elements of an integral domain which are not product of irreducible elements.

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#ring-theory #integral-domain #unique-factorization-domains
81
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Is $\mathbb{Z} [(-2)^{\frac{1}{n}}]$ ever a UFD?

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#abstract-algebra #ring-theory #factoring #unique-factorization-domains
170
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Prove $ \mathbb{Z}[\sqrt {-5}] $ is not a UFD without factoring?

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#ring-theory #factoring #alternative-proof #unique-factorization-domains
157
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In a UFD, why does $ab | c^n$ implies $ab | c$?

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#unique-factorization-domains
382
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Ideal in polynomial ring which contains no non-zero prime ideal

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#algebraic-geometry #commutative-algebra #noetherian #unique-factorization-domains #krull-dimension

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