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 ${\displaystyle ab\in {\mathfrak {p}}}$ ${\displaystyle a\in {\mathfrak {p}}}$ ${\displaystyle b\in {\mathfrak {p}}.}$

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

• Every fully prime ideal is prime, but not the other way around. For example, the zero ideal in the ring of real matrices is ${\displaystyle 2\times 2}$ prime, but not fully prime.
• In commutative rings, prime and fully prime are equivalent.

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 ${\displaystyle S}$ ${\displaystyle m<n}$ ${\displaystyle {\mathfrak {q}}_{i}}$ ${\displaystyle {\mathfrak {p}}_{i}}$ ${\displaystyle 1\leq i\leq m}$

${\displaystyle {\mathfrak {q}}_{1}\supseteq {\mathfrak {q}}_{2}\supseteq \cdots \supseteq {\mathfrak {q}}_{n}}$

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