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 :
- From follows or
In addition is complete prime ideal or vollprim if is genuine and if all the following applies:
- From follows or
Equivalent Definitions
- A two-sided ideal is prime if and only if it is real and if applies to all :
- OFF (for all true ) follows ( or ).
- A two-sided ideal is fully prime if and only if it is real and if the factor ring is zero divisor .
spectrum
The set of all (real) prime ideals of a ring is called the spectrum of and is noted with .
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 prime, but not fully prime.
- In commutative rings, prime and fully prime are equivalent.
In commutative rings with one element, the following applies:
- An element is a prime element if and only if the main ideal generated by is a prime ideal.
- An ideal is prime if and only if the factor ring is an integrity ring .
- If a prime ideal contains an intersection of finitely many ideals of , then it also contains one of the ideals .
- 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.
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.
- 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.
- In the ring , the maximum ideal is not a prime ideal.
- A maximal ideal of a ring is prime if and only if . In particular, is prime if contains a one element.
- The zero ideal in a commutative ring is a prime ideal if and only if there is an integrity domain .
- 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.
- .
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.
Furthermore, the going-down property fulfills if the following applies: is
a chain of prime ideals in and
a chain of prime ideals in with , so that moreover lies over for all , so the latter can be made into a chain
complement so that each one lies above . This is fulfilled, among other things, if there are integrity rings and are completely closed.
Individual evidence
- ↑ 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 '
- ^ K. Meyberg: Algebra, part 1 , Carl Hanser Verlag Munich (1975), ISBN = 3-446-11965-5, sentence 3.6.5
- ↑ 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