# Prime ideal

In ring theory , a prime ideal is a subset of a ring that behaves like a prime number as an element of the whole numbers.

## Definitions

Let it be a ring . Then a two-sided ideal is called a prime ideal or prime , if it is real , that is , and if the following applies to all ideals : ${\ displaystyle R}$ ${\ displaystyle {\ mathfrak {p}} \ subseteq R}$ ${\ displaystyle {\ mathfrak {p}}}$ ${\ displaystyle {\ mathfrak {p}} \ neq R}$ ${\ displaystyle {\ mathfrak {a, b}} \ subseteq R}$ From follows or${\ displaystyle {\ mathfrak {ab}} \ subseteq {\ mathfrak {p}}}$ ${\ displaystyle {\ mathfrak {a}} \ subseteq {\ mathfrak {p}}}$ ${\ displaystyle {\ mathfrak {b}} \ subseteq {\ mathfrak {p}}.}$ In addition is complete prime ideal or vollprim if is genuine and if all the following applies: ${\ displaystyle {\ mathfrak {p}}}$ ${\ displaystyle {\ mathfrak {p}}}$ ${\ displaystyle a, b \ in R}$ From follows or${\ displaystyle from \ in {\ mathfrak {p}}}$ ${\ displaystyle a \ in {\ mathfrak {p}}}$ ${\ displaystyle b \ in {\ mathfrak {p}}.}$ ### Equivalent Definitions

• A two-sided ideal is prime if and only if it is real and if applies to all :${\ displaystyle {\ mathfrak {p}} \ subseteq R}$ ${\ displaystyle a, b \ in R}$ OFF (for all true ) follows ( or ).${\ displaystyle r \ in R}$ ${\ displaystyle arb \ in {\ mathfrak {p}}}$ ${\ displaystyle a \ in {\ mathfrak {p}}}$ ${\ displaystyle b \ in {\ mathfrak {p}}}$ • A two-sided ideal is fully prime if and only if it is real and if the factor ring is zero divisor .${\ displaystyle {\ mathfrak {p}} \ subseteq R}$ ${\ displaystyle R / {\ mathfrak {p}}}$ ### spectrum

The set of all (real) prime ideals of a ring is called the spectrum of and is noted with . ${\ displaystyle R}$ ${\ displaystyle R}$ ${\ displaystyle \ mathrm {Spec} (R)}$ ## properties

In commutative rings with one element, the following applies: ${\ displaystyle R}$ • An element is a prime element if and only if the main ideal generated by is a prime ideal.${\ displaystyle p \ in R \ backslash \ left \ {0 \ right \}}$ ${\ displaystyle p}$ ${\ displaystyle (p)}$ • An ideal is prime if and only if the factor ring is an integrity ring .${\ displaystyle {\ mathfrak {p}} \ subset R}$ ${\ displaystyle R / {\ mathfrak {p}}}$ • If a prime ideal contains an intersection of finitely many ideals of , then it also contains one of the ideals .${\ displaystyle {\ mathfrak {a}} _ {1} \ cap \ ldots \ cap {\ mathfrak {a}} _ {n}}$ ${\ displaystyle R}$ ${\ displaystyle {\ mathfrak {a}} _ {i}}$ • An ideal is a prime ideal if and only if the complementary set is multiplicatively closed. This leads to the concept of localization of , by which is the ring understands that one as writes.${\ displaystyle {\ mathfrak {p}} \ subset R}$ ${\ displaystyle S = R \ setminus {\ mathfrak {p}}}$ ${\ displaystyle {\ mathfrak {p}}}$ ${\ displaystyle S ^ {- 1} R}$ ${\ displaystyle R _ {\ mathfrak {p}}}$ ## Examples

• The set of even whole numbers is a prime ideal in the ring of whole numbers , since a product of two whole numbers is only even if at least one factor is even.${\ displaystyle 2 \ mathbb {Z}}$ ${\ displaystyle \ mathbb {Z}}$ • The set of integers divisible by 6 is not a prime ideal in , since 2 3 = 6 lies in the subset, but neither 2 nor 3.${\ displaystyle 6 \ mathbb {Z}}$ ${\ displaystyle \ mathbb {Z}}$ • In the ring , the maximum ideal is not a prime ideal.${\ displaystyle R = 2 \ mathbb {Z}}$ ${\ displaystyle {\ mathfrak {m}} = 4 \ mathbb {Z}}$ • A maximal ideal of a ring is prime if and only if . In particular, is prime if contains a one element.${\ displaystyle {\ mathfrak {m}} \ subseteq R}$ ${\ displaystyle R}$ ${\ displaystyle RR \ nsubseteq {\ mathfrak {m}}}$ ${\ displaystyle {\ mathfrak {m}}}$ ${\ displaystyle R}$ • The zero ideal in a commutative ring is a prime ideal if and only if there is an integrity domain .${\ displaystyle (0) \ subset R}$ ${\ displaystyle R}$ ${\ displaystyle R}$ • In a non-commutative ring, this equivalence does not hold.
• In general, the archetype of a prime ideal under a ring homomorphism is a prime ideal.

## Lying Over and Going Down

In the following there is always a commutative ring and a whole ring expansion . Then for each prime ideal a prime ideal , so that over lies , d. H. ${\ displaystyle R}$ ${\ displaystyle R \ subset S}$ ${\ displaystyle {\ mathfrak {p}} \ subset R}$ ${\ displaystyle {\ mathfrak {q}} \ subset S}$ ${\ displaystyle {\ mathfrak {q}}}$ ${\ displaystyle {\ mathfrak {p}}}$ ${\ displaystyle {\ mathfrak {p}} = {\ mathfrak {q}} \ cap R}$ .

In this case it is also said that the Lying Over property is fulfilled. Is also an embedding in , the of is induced map with surjective. ${\ displaystyle S / R}$ ${\ displaystyle f: R \ hookrightarrow S}$ ${\ displaystyle R}$ ${\ displaystyle S}$ ${\ displaystyle f}$ ${\ displaystyle f ^ {*}: \ mathrm {Spec} (S) \ longrightarrow \ mathrm {Spec} (R)}$ ${\ displaystyle {\ mathfrak {q}} \ longmapsto f ^ {- 1} ({\ mathfrak {q}})}$ Furthermore, the going-down property fulfills if the following applies: is ${\ displaystyle S / R}$ ${\ displaystyle {\ mathfrak {p}} _ {1} \ supseteq {\ mathfrak {p}} _ {2} \ supseteq \ cdots \ supseteq {\ mathfrak {p}} _ {n}}$ a chain of prime ideals in and ${\ displaystyle R}$ ${\ displaystyle {\ mathfrak {q}} _ {1} \ supseteq {\ mathfrak {q}} _ {2} \ supseteq \ cdots \ supseteq {\ mathfrak {q}} _ {m}}$ a chain of prime ideals in with , so that moreover lies over for all , so the latter can be made into a chain ${\ displaystyle S}$ ${\ displaystyle m ${\ displaystyle {\ mathfrak {q}} _ {i}}$ ${\ displaystyle {\ mathfrak {p}} _ {i}}$ ${\ displaystyle 1 \ leq i \ leq m}$ ${\ displaystyle {\ mathfrak {q}} _ {1} \ supseteq {\ mathfrak {q}} _ {2} \ supseteq \ cdots \ supseteq {\ mathfrak {q}} _ {n}}$ complement so that each one lies above . This is fulfilled, among other things, if there are integrity rings and are completely closed. ${\ displaystyle {\ mathfrak {q}} _ {i}}$ ${\ displaystyle {\ mathfrak {p}} _ {i}}$ ${\ displaystyle R, S}$ ${\ displaystyle R}$ ## Individual evidence

1. Louis H. Rowen: Ring Theory. Volume 1. Academic Press Inc., Boston et al. 1988, ISBN 0-125-99841-4 ( Pure and Applied Mathematics 127), definition 2.2.3 '
2. ^ K. Meyberg: Algebra, part 1 , Carl Hanser Verlag Munich (1975), ISBN = 3-446-11965-5, sentence 3.6.5
3. Ernst Kunz: Introduction to Commutative Algebra and Algebraic Geometry , Vieweg (1980), ISBN 3-528-07246-6 , Chapter III, §4, example d) behind theorem 3.5