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More generally, a Prüfer ring is a commutative ring in which every non-zero finitely generated ideal containing a non-zero-divisor is invertible (that is, projective). A commutative ring is said to be arithmetical if for every maximal ideal m in R, the localization R m of R at m is a chain ring. With this definition, a Prüfer domain is an ...
A ring R is prime if and only if the zero ideal {0} is a prime ideal in the noncommutative sense. This being the case, the equivalent conditions for prime ideals yield the following equivalent conditions for R to be a prime ring: For any two ideals A and B of R, AB = {0} implies A = {0} or B = {0}.
The ring = of algebraic integers in a number field K is Noetherian, integrally closed, and of dimension one: to see the last property, observe that for any nonzero prime ideal I of R, R/I is a finite set, and recall that a finite integral domain is a field; so by (DD4) R is a Dedekind domain. As above, this includes all the examples considered ...
Any field is a Jacobson ring. Any principal ideal domain or Dedekind domain with Jacobson radical zero is a Jacobson ring. In principal ideal domains and Dedekind domains, the nonzero prime ideals are already maximal, so the only thing to check is if the zero ideal is an intersection of maximal ideals.
Another important example of a DVR is the ring of formal power series = [[]] in one variable over some field .The "unique" irreducible element is , the maximal ideal of is the principal ideal generated by , and the valuation assigns to each power series the index (i.e. degree) of the first non-zero coefficient.
In a commutative ring with unity, every maximal ideal is a prime ideal. The converse is not always true: for example, in any nonfield integral domain the zero ideal is a prime ideal which is not maximal. Commutative rings in which prime ideals are maximal are known as zero-dimensional rings, where the dimension used is the Krull dimension.
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This extends the definition for commutative rings. 4. prime ring : A nonzero ring R is called a prime ring if for any two elements a and b of R with aRb = 0, we have either a = 0 or b = 0. This is equivalent to saying that the zero ideal is a prime ideal (in the noncommutative sense.) Every simple ring and every domain is a prime ring. primitive 1.