quotient ring modulo prime ideal
Proof. . First, let be a prime ideal of . Then is of course a commutative ring and has the unity . If the product of two residue classes vanishes, i.e. equals , then we have , and therefore must belong to . Since is , either or belongs to , i.e. or . Accordingly, has no zero divisors and is an integral domain.
. Conversely, let be an integral domain and let the product of two elements of belong to . It follows that . Since has no zero divisors, or . Thus, or belongs to , i.e. is a prime ideal.
|Title||quotient ring modulo prime ideal|
|Date of creation||2013-03-22 17:37:09|
|Last modified on||2013-03-22 17:37:09|
|Last modified by||pahio (2872)|