Editing Prime number (section)

=== Prime elements of a ring ===
{{Main|Prime element|Irreducible element}}
[[File:Gaussian primes.svg|thumb|All [[Gaussian prime]]s with norm squared less than 500]]
A [[commutative ring]] is an [[algebraic structure]] where addition, subtraction and multiplication are defined. The integers are a ring, and the prime numbers in the integers have been generalized to rings in two different ways, ''prime elements'' and ''irreducible elements''. An element {{tmath|p}} of a ring {{tmath|R}} is called prime if it is nonzero, has no [[multiplicative inverse]] (that is, it is not a [[Unit (ring theory)|unit]]), and satisfies the following requirement: whenever {{tmath|p}} divides the product <math>xy</math> of two elements of {{tmath|R}}, it also divides at least one of {{tmath|x}} or {{tmath|y}}. An element is irreducible if it is neither a unit nor the product of two other non-unit elements. In the ring of integers, the prime and irreducible elements form the same set,
: <math>\{ \dots, -11, -7, -5, -3, -2, 2, 3, 5, 7, 11, \dots \}\, .</math>
In an arbitrary ring, all prime elements are irreducible. The converse does not hold in general, but does hold for [[unique factorization domain]]s.<ref>{{cite book | last = Lauritzen | first = Niels | doi = 10.1017/CBO9780511804229 | isbn = 978-0-521-53410-9 | location = Cambridge | mr = 2014325 | page = 127 | publisher = Cambridge University Press | title = Concrete Abstract Algebra: From numbers to Gröbner bases | url = https://books.google.com/books?id=BdAbcje-TZUC&pg=PA127 | year = 2003 }}</ref>

The fundamental theorem of arithmetic continues to hold (by definition) in unique factorization domains. An example of such a domain is the [[Gaussian integer]]s {{tmath|\mathbb{Z}[i]}}, the ring of [[complex number]]s of the form <math>a+bi</math> where {{tmath|i}} denotes the [[imaginary unit]] and {{tmath|a}} and {{tmath|b}} are arbitrary integers. Its prime elements are known as [[Gaussian prime]]s. Not every number that is prime among the integers remains prime in the Gaussian integers; for instance, the number 2 can be written as a product of the two Gaussian primes <math>1+i</math> and {{tmath|1-i}}. Rational primes (the prime elements in the integers) congruent to 3 mod 4 are Gaussian primes, but rational primes congruent to 1 mod 4 are not.<ref>{{harvnb|Lauritzen|2003}}, Corollary 3.5.14, p. 133; Lemma 3.5.18, p. 136.</ref> This is a consequence of [[Fermat's theorem on sums of two squares]],
which states that an odd prime {{tmath|p}} is expressible as the sum of two squares, {{tmath|1= p=x^2+y^2 }}, and therefore factorable as {{tmath|1= p=(x+iy)(x-iy) }}, exactly when {{tmath|p}} is 1 mod 4.<ref>{{harvnb|Kraft|Washington|2014}}, [https://books.google.com/books?id=4NAqBgAAQBAJ&pg=PA297 Section 12.1, Sums of two squares, pp. 297–301].</ref>