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Everything about Unique factorization domain
Type of integral domain
In mathematics, a unique factorization domain (UFD) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is an integral domain in which every non-zero non-unit element can be written as a product of irreducible elements, uniquely up to order and units. Wikipedia →
🧭 Start here: a 6-step reading path
- Read the overview: Unique factorization domainA 5-minute orientation
- Skim a recent paper: Smith form of the matrix over the unique factorization domainInfluential work since 2022
- Pick up a classic book: Advanced number theoryThe most-reprinted book on the subject
- Read the landmark paper: Noncommutative unique factorization domainsCited 50 times
- Go sideways: Principal ideal domainA closely related topic
- Go sideways: Integral domainA closely related topic
📖 Definitions
Noun A unique factorization ring which is also an integral domain.
🗂️ The subject at a glance
🏆 Landmark papers (most cited)
Noncommutative unique factorization domains
On unique factorization domains
Unique Factorization Domains
Non-commutative unique factorization domains
On unique factorization domains
Infinite primes and unique factorization in a principal right ideal domain
Three-dimensional unique factorization domain which is not catenary
AP-domains and unique factorization
🆕 Influential since 2022
Smith form of the matrix over the unique factorization domain
Generalizations of Samuel's criteria for a ring to be a unique factorization domain
The extension property in a category of torsion modules over a unique factorization domain
Relation Between Integral Domain and Unique Factorization Domain
“A brief study of Unique factorization domain in Integral Domain”
📚 Books worth reading
Advanced number theoryHarvey Cohn · 1980
Ranked by number of editions on Open Library, a rough measure of how lasting a book is.
🔗 Related topics
- Principal ideal domain In mathematics, a principal ideal domain, or PID, is an integral domain (that is, a non-zero commutative ring without nonzero zero divisors) in which every…
- Integral domain In mathematics, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. In an integral domain, every…
- Gauss's lemma (polynomials) In algebra, Gauss's lemma, named after Carl Friedrich Gauss, is a theorem about polynomials over the integers, or, more generally, over a unique factorization…
- Bézout domain In mathematics, a Bézout domain is an integral domain in which the sum of two principal ideals is also a principal ideal. This means that Bézout's identity…
- Factorization In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object…
- GCD domain In mathematics, a GCD domain is an integral domain R with the property that any two elements have a greatest common divisor (GCD); i.e., there is a minimum…
- Atomic domain In mathematics, more specifically ring theory, an atomic domain or factorization domain is an integral domain in which every non-zero non-unit can be written…
- Prime element In mathematics, specifically in abstract algebra, a prime element of a commutative ring is an object satisfying certain properties similar to the prime…