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A consequence of Gauss' lemma

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#abstract-algebra #polynomials #irreducible-polynomials #unique-factorization-domains
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$R$ UFD and integral domain $\Rightarrow$ every $a\neq 0 \in R\setminus R^\times$ is a product of prime elements

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#abstract-algebra #ring-theory #integral-domain #unique-factorization-domains
279
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Prove that the quotient ring is not a PID

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#commutative-algebra #algebraic-number-theory #principal-ideal-domains #unique-factorization-domains
244
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A Noetherian domain $R$ is a UFD iff $I_{fg}=\{h\in R \mid hg\in(f)\}$ is principal for all $f,g\in R$.

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#abstract-algebra #ring-theory #noetherian #unique-factorization-domains
796
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Question about proving $\mathbb{Z}[i]$ is not UFD

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#abstract-algebra #ring-theory #unique-factorization-domains
47
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Proof of a coprimes property in a UFD

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#abstract-algebra #unique-factorization-domains
155
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Is there a quotient of a symmetric algebra which is in bijection with the simple tensors?

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#abstract-algebra #commutative-algebra #tensor-products #unique-factorization-domains
239
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Prime ideals in $\mathbb{Z}[X], \mathbb{Q}[X], \mathbb{F}_{19}[X]$.

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#abstract-algebra #ring-theory #maximal-and-prime-ideals #unique-factorization-domains #polynomial-rings
120
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Let R be UFD. Consider $R[[x]]$

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#abstract-algebra #ring-theory #unique-factorization-domains
57
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What does addition contribute to the concept of unique factorisation in rings?

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#abstract-algebra #group-theory #ring-theory #factoring #unique-factorization-domains
118
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Unique Factorization Theorem for Integral Domains

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#abstract-algebra #ring-theory #unique-factorization-domains
783
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Subring of a UFD need not be a UFD

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#abstract-algebra #ring-theory #commutative-algebra #unique-factorization-domains
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Quotient of UFD

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#abstract-algebra #ring-theory #unique-factorization-domains
380
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G.C.D. of two elements in a U.F.D.

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#abstract-algebra #unique-factorization-domains
162
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Visualizing euclidean rings

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#abstract-algebra #ring-theory #unique-factorization-domains #euclidean-domain

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