Prime ideal

In theory, the ring is a prime ideal is a subset of a ring, similar to acts as a prime number as the element of the whole numbers.

Definitions

It was a ring. Then is called a two-sided ideal prime ideal or prime, if is real, so, and if for all ideals:

Also, is called complete prime ideal or vollprim if is real and if for all:

A two-sided ideal if and only vollprim, if it is genuine and if the factor ring is zero divisors.

The set of all (real) prime ideals of a ring is called spectrum and with Spec ( ) listed.

Each fully- prime ideal is prime, but not vice versa. For example, the zero ideal in the ring of real matrices prim, but not vollprim.
In commutative rings are prim and vollprim equivalent.

In commutative rings with identity holds:

An element is a prime element if and only if it is generated by principal ideal is a prime ideal.
An ideal is prime if and only if the factor ring is an integral domain.
Contains a prime ideal an average of a finite number of ideals, it also contains one of the ideals.
An ideal is a prime ideal if and only if the complementary set is complete multiplicative. This leads to the concept of localization, which is meant the ring that you as writes.

The set of even integers being a prime ideal in the ring of integers as a product of two integers is just only if at least one factor is straight.
The set of integers divisible by 6 is not a prime ideal, since 2.3 = 6 is in the subset, but neither 2 nor 3
In the ring, the maximal ideal is not a prime ideal.
A maximal ideal of a ring if and only prime if. In particular, is prime if a unit element contains.
The zero ideal in a commutative ring is a prime ideal if and only if an integral domain is.
In a non- commutative ring this equivalence does not hold, such as the example of the n × n- matrices is for n> 1.
In general, the inverse image of a prime ideal is a prime ideal under a ring homomorphism

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