About the localization of a UFD - Mathematics Stack Exchange
Dec 17, 2014 · I was wondering, is the localization of a UFD also a UFD? How would one go about proving this? It seems like it would be kind of messy to prove if it is true. If it is not true, …
Understanding the definition of Unique Factorization Domains
Oct 16, 2015 · The point of a UFD is that any element can be rewritten as a product of irreducible factors, where any other product of irreducible factors is just a rearrangement of the exact …
Is every PID a UFD? - Mathematics Stack Exchange
Jan 31, 2020 · So we have the following characterization of UFDs: $$\text {UFD}\iff \text {Atomic domain}+\text {AP-domain}.$$ Therefore you can prove that a PID is a UFD showing that is …
Integral closure and rational root test over a UFD
Dec 28, 2021 · However, in a UFD it is common to prove the Law and Lemma using prime factorizations - which yields less generality and, further, is less constructive - since in many …
Is $\\mathbb Z[ \\sqrt{-3}]$ UFD? - Mathematics Stack Exchange
This is norm-Euclidean and in particular a UFD. Note that all UFDs are integrally closed, so this is another way to see that $\mathbf {Z} [\sqrt {-3}]$ does not have unique factorization.
abstract algebra - Does $A$ a UFD imply that $A [T]$ is also a UFD ...
The fact that $A$ is a UFD implies that $A [X]$ is a UFD is very standard and can be found in any textbook on Algebra (for example, it is Proposition 2.9.5 in these notes by Robert Ash).
abstract algebra - Prove that a UFD is a PID if and only if every ...
Prove that a UFD is a PID if and only if every nonzero prime ideal is maximal. The forward direction is standard, and the reverse direction is giving me trouble. In particular, I can prove …
Is UFD an integral domain? - Mathematics Stack Exchange
Nov 24, 2020 · Being an integral domain is part of the definition of a UFD; this is what the "domain" part of "unique factorization domain" refers to. Consider for instance $\mathbb …
abstract algebra - Hilbert basis theorem implies $F [x, y]$ is a UFD ...
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The definition of a unique factorisation domain
May 14, 2012 · The definition of a UFD involves two parts, existence and uniqueness of factorizations into atoms (irreducibles). Your quote mistakenly omits uniqueness. More …