Prime Ideals For Noncommutative Rings
The notion of a prime ideal can be generalized to noncommutative rings by using the commutative definition "ideal-wise". Wolfgang Krull advanced this idea in 1928. The following content can be found in texts such as (Goodearl 2004) and (Lam 2001). If R is a (possibly noncommutative) ring and P is an ideal in R other than R itself, we say that P is prime if for any two ideals A and B of R:
- If the product of ideals is contained in, then at least one of and is contained in .
It can be shown that this definition is equivalent to the commutative one in commutative rings. It is readily verified that if an ideal of a noncommutative ring R satisfies the commutative definition of prime, then it also satisfies the noncommutative version. An ideal P satisfying the commutative definition of prime is sometimes called a completely prime ideal to distinguish it from other merely prime ideals in the ring. Completely prime ideals are prime ideals, but the converse is not true. For example, the zero ideal in the ring of n × n matrices over a field is a prime ideal, but it is not completely prime.
This is close to the historical point of view of ideals as ideal numbers, as for the ring Z "A is contained in P" is another way of saying "P divides A", and the unit ideal R represents unity.
Equivalent formulations of the ideal P≠R being prime include the following properties:
- For all a and b in R, (a)(b)⊆P implies a∈P or b∈P.
- For any two right ideals of R, AB⊆P implies A⊆P or B⊆P.
- For any two left ideals of R, AB⊆P implies A⊆P or B⊆P.
- For any elements a and b of R, if aRb⊆P, then a∈P or b∈P.
Prime ideals in commutative rings are characterized by having multiplicatively closed complements in R, and with slight modification, a similar characterization can be formulated for prime ideals in noncommutative rings. A nonempty subset S⊆R is called an m-system if for any a and b in S, there exists r in R such that arb is in S. The following item can then be added to the list of equivalent conditions above:
- The complement R\P is an m-system.
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