Let M, N be two finitely generated modules over a commutative ring, such that Ann ( M ) + Ann ( N ) is a proper ideal.
2.
A field is a commutative ring in which there are no nontrivial proper ideals, so that any field is a Dedekind domain, however in a rather vacuous way.
3.
More specifically, the truth of the idea that any proper ideal in a ring is contained in at least one ( proper ) maximal ideal requires the application of Zorn's lemma.
4.
In fact, the mapping I \ mapsto \ Gamma _ I defines an inclusion-reversing bijection between the set of proper ideals of " D " and the set of segments of \ Gamma.
5.
Then its coefficients generate a proper ideal " I ", which by Krull's theorem ( which depends on the axiom of choice ) is contained in a maximal ideal " m " of " R ".
6.
A proper ideal " P " of " R " is called a prime ideal if for any elements x, y \ in R we have that xy \ in P implies either x \ in P or y \ in P.
7.
Principal right ideal rings and right B�zout rings are also closed under quotients, that is, if " I " is a proper ideal of principal right ideal ring " R ", then the quotient ring " R / I " is also principal right ideal ring.
8.
It was shown that while rings of algebraic integers do not always have unique factorization into primes ( because they need not be principal ideal domains ), they do have the property that every proper ideal admits a unique factorization as a product of prime ideals ( that is, every ring of algebraic integers is a Dedekind domain ).
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