Integrity ring

In algebra , an integrity ring or area of integrity is a commutative ring with a unity that is different from the zero ring and is free of zero divisors .

Alternatively, one can define an integrity ring as a commutative ring with 1, in which the zero ideal is a prime ideal , or as a partial ring of a body . There is also a weakened definition in which no one element is required, only that there is at least one non-zero element in the ring. However, many sentences about integrity rings require a one, so this property is usually included in the definition. ${\ displaystyle \ lbrace 0 \ rbrace}$

Examples

• The best known example is the ring of whole numbers.${\ displaystyle \ mathbb {Z}}$
• Every body is a ring of integrity. Conversely, every Artinian integrity ring is a body. In particular, every finite integrity ring is a finite field : it is easy to verify that the mapping is injective for one . Since is finite, the bijectivity of . So there exists a unique element of , so . Since any other than zero was chosen, it follows that each has an inverse in , that is, that it is a body.${\ displaystyle a \ in R \ setminus \ {0 \}}$${\ displaystyle \ varphi: R \ longrightarrow R, r \ mapsto ar}$${\ displaystyle R}$${\ displaystyle \ varphi}$${\ displaystyle b}$${\ displaystyle R}$${\ displaystyle 1 = \ varphi (b) = from}$${\ displaystyle a}$${\ displaystyle a}$${\ displaystyle R}$${\ displaystyle R}$
• A polynomial ring is an integrity ring if and only if the coefficients come from an integrity ring. For example, the ring of polynomials with integer coefficients is an integrity ring, as is the ring of real polynomials in two variables.${\ displaystyle \ mathbb {Z} [X]}$${\ displaystyle \ mathbb {R} [X, Y]}$
• The ring of all real numbers of the form with integers is an integrity ring because it is a partial ring of . In general, the whole ring of an algebraic number field is always an integrity ring.${\ displaystyle a + b {\ sqrt {2}}}$${\ displaystyle a, b}$${\ displaystyle \ mathbb {R}}$
• Is a commutative ring with 1, so is factor ring if and only an integrity ring when a prime ideal in is. The remainder class ring is an integrity ring (for ) if and only if is a prime number .${\ displaystyle R}$ ${\ displaystyle R / P}$${\ displaystyle P}$${\ displaystyle R}$ ${\ displaystyle \ mathbb {Z} / n \ mathbb {Z}}$${\ displaystyle n> 0}$${\ displaystyle n}$
• If there is a domain (a connected open subset) in the complex numbers , the ring of holomorphic functions is an integrity ring.${\ displaystyle U \ subseteq \ mathbb {C}}$${\ displaystyle \ operatorname {H} (U)}$ ${\ displaystyle f \ colon U \ to \ mathbb {C}}$
• For an integrity ring and a natural number , the matrix ring is an integrity ring if and only if applies.${\ displaystyle R}$${\ displaystyle n}$ ${\ displaystyle R ^ {n \ times n}}$${\ displaystyle n = 1}$

Divisibility

If and are elements of the integrity ring , then one calls a divisor of and a multiple of (and also says: divides ), if there is an element in such that . Then you write , otherwise . ${\ displaystyle a}$${\ displaystyle b}$${\ displaystyle R}$${\ displaystyle a}$${\ displaystyle b}$${\ displaystyle b}$${\ displaystyle a}$${\ displaystyle a}$ ${\ displaystyle b}$${\ displaystyle x}$${\ displaystyle R}$${\ displaystyle ax = b}$${\ displaystyle a \ mid b}$${\ displaystyle a \ nmid b}$

The following rules of divisibility apply:

• If and , then it follows .${\ displaystyle a \ mid b}$${\ displaystyle b \ mid c}$${\ displaystyle a \ mid c}$
• Applies , then also applies to everyone , especially also .${\ displaystyle a \ mid b}$${\ displaystyle a \ mid bc}$${\ displaystyle c \ in R}$${\ displaystyle a \ mid -b}$
• Applies and then applies and .${\ displaystyle a \ mid b}$${\ displaystyle a \ mid c}$${\ displaystyle a \ mid b + c}$${\ displaystyle a \ mid b {-} c}$

The first rule is that divisibility is transitive . The second and third rule say that the set of multiples of an element forms an ideal in ; this is also noted as. ${\ displaystyle aR}$${\ displaystyle a}$${\ displaystyle R}$${\ displaystyle (a)}$

units

Ring elements that divide the figure 1 are called units of . The units are identical to the invertible elements and share all other elements. The set of units of is denoted by and together with the ring multiplication forms an Abelian group - the so-called unit group of . A ring element that is not a unit is called a non-unit . ${\ displaystyle R}$${\ displaystyle R}$${\ displaystyle R ^ {*}}$${\ displaystyle R}$

A finite subgroup of the unit group of an integrity ring is always cyclic . This statement becomes wrong if one waives the freedom from zero divisors. The unit group is finite and not cyclical by itself. The statement becomes just as wrong if one maintains the zero divisor freedom, but renounces the commutativity: The quaternion group is a finite subgroup of the unit group of the zero divisor -free, but not commutative ring of quaternions and is not cyclic. ${\ displaystyle \ mathbb {Z} / 8 \ mathbb {Z}}$ ${\ displaystyle Q_ {8}}$ ${\ displaystyle \ mathbb {H}}$

Associated elements

Apply and , then called and associated with each other . Two ring elements and are associated if and only if there is a unity such that . ${\ displaystyle a \ mid b}$${\ displaystyle b \ mid a}$${\ displaystyle a}$${\ displaystyle b}$${\ displaystyle a}$${\ displaystyle b}$${\ displaystyle u}$${\ displaystyle au = b}$

Irreducibility

An element is called reducible if it is a unit or a product of two (not necessarily different) non-units, otherwise it is called irreducible .

Prime elements

An element is called a prime element (or prime for short ) if it is neither 0 nor a unit and the following also applies: From follows or . The main ideal is then a prime ideal . Conversely, if the main ideal of a nonunity different from zero is a prime ideal, then it is prime. (The zero ideal is a prime ideal in integrity rings, the main ideals of units are already the entire ring.) ${\ displaystyle p}$${\ displaystyle p}$${\ displaystyle p \ mid ab}$${\ displaystyle p \ mid a}$${\ displaystyle p \ mid b}$ ${\ displaystyle (p)}$${\ displaystyle (p)}$${\ displaystyle p}$${\ displaystyle p}$

Relationship between prime and irreducible elements

Every prime element is irreducible (for this statement the zero divisor of the ring is required), but every irreducible element is not always prime. In the ring are , , and irreducible but not prime: shares, for example, not yet , but their product. ${\ displaystyle \ mathbb {Z} [{\ sqrt {-5}}]}$${\ displaystyle 2}$${\ displaystyle 3}$${\ displaystyle 1 + {\ sqrt {-5}}}$${\ displaystyle 1 - {\ sqrt {-5}}}$${\ displaystyle 3}$${\ displaystyle 1 + {\ sqrt {-5}}}$${\ displaystyle 1 - {\ sqrt {-5}}}$

In main ideal rings and more generally in factorial rings, however, both terms are identical. Thus, in the primes usually only as a positive, irreducible elements defined. However, these elements are also prime elements, since factorial and therefore every irreducible element is prime. However, there are also the negative counterparts of the prim numbers, prim elements , which shows that the concept of the prime element is more general than the concept of the prime number. ${\ displaystyle \ mathbb {Z}}$${\ displaystyle \ mathbb {Z}}$${\ displaystyle \ mathbb {Z}}$

Quotient field

If there is an integrity ring, then there is a smallest body that contains as a partial ring. The body is uniquely determined except for isomorphism and is called the quotient body of . Its elements have the form with The quotient field is an example of a construction with an integrity ring, in which no one (in the definition of the integrity ring) is required, only some non-zero element. ${\ displaystyle R}$ ${\ displaystyle \ operatorname {Quot} (R),}$${\ displaystyle R}$${\ displaystyle \ operatorname {Quot} (R) \, \!}$${\ displaystyle R}$${\ displaystyle {\ tfrac {a} {b}}}$${\ displaystyle a, b \ in R, b \ neq 0.}$

The quotient field of the ring of integers is the field of rational numbers . The quotient field of a body is the body itself.

Alternatively, one can construct quotient fields via localizations of according to the zero ideal . ${\ displaystyle R}$${\ displaystyle \ lbrace 0 \ rbrace}$

In the abstract, quotient bodies are defined by the following universal property :

A quotient of a ring body , a pair of a field K and a homomorphism from according with the property that for every body with homomorphism exactly one Körperhomomorphismus with exist.${\ displaystyle R}$${\ displaystyle (K, \ phi)}$${\ displaystyle \ phi}$${\ displaystyle R}$${\ displaystyle K}$${\ displaystyle L}$${\ displaystyle \ psi \ colon R \ to L}$${\ displaystyle \ alpha \ colon K \ to L}$${\ displaystyle \ psi = \ alpha \ circ \ phi}$

Characteristic

The characteristic of an integrity ring is either 0 or a prime number, because if a ring has a characteristic , then it follows (due to the fact that it is not divided by zero) either or . But this is already the definition of the characteristic (smallest with ), which is why either is or and is therefore prime. Note that an integrity ring (more precisely: the commutativity of a ring) is not absolutely necessary for this proof, a zero-divisor-free ring with 1 is sufficient. ${\ displaystyle c = k \ cdot l}$${\ displaystyle \ sum _ {i = 1} ^ {c} 1 = \ sum _ {i = 1} ^ {k \ cdot l} 1 = \ left (\ sum _ {i = 1} ^ {k} 1 \ right) \ cdot \ left (\ sum _ {i = 1} ^ {l} 1 \ right) = 0}$${\ displaystyle \ sum _ {i = 1} ^ {k} 1 = 0}$${\ displaystyle \ sum _ {i = 1} ^ {l} 1 = 0}$${\ displaystyle n}$${\ displaystyle \ sum _ {i = 1} ^ {n} 1 = 0}$${\ displaystyle k = c}$${\ displaystyle l = c}$${\ displaystyle c}$

If an integrity ring with the prime number characteristic , then the mapping is an injective ring homomorphism and is called Frobenius homomorphism . If the considered ring is finite, it is even bijective, i.e. an automorphism. ${\ displaystyle R}$${\ displaystyle p}$${\ displaystyle f \ colon R \ to R, \; x \ mapsto x ^ {p}}$${\ displaystyle f}$

Individual evidence

1. ^ André Weil Basic number theory , Springer-Verlag 1995