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If F is a field, then the only maximal ideal is {0}. In the ring Z of integers, the maximal ideals are the principal ideals generated by a prime number. More generally, all nonzero prime ideals are maximal in a principal ideal domain. The ideal (,) is a maximal ideal in ring [].
An integral domain where every finitely generated ideal is principal (that is, a Bézout domain) satisfies (ACCP) if and only if it is a principal ideal domain. [4] The ring Z+XQ[X] of all rational polynomials with integral constant term is an example of an integral domain (actually a GCD domain) that does not satisfy (ACCP), for the chain of ...
In principal ideal domains a near converse holds: every nonzero prime ideal is maximal. All principal ideal domains are integrally closed. The previous three statements give the definition of a Dedekind domain, and hence every principal ideal domain is a Dedekind domain. Let A be an integral domain, the following are equivalent. A is a PID.
The ideal is contained inside the ideal , since every multiple of is also a multiple of . In turn, the ideal J {\displaystyle J} is contained in the ideal Z {\displaystyle \mathbb {Z} } , since every multiple of 2 {\displaystyle 2} is a multiple of 1 {\displaystyle 1} .
In any ring R, a maximal ideal is an ideal M that is maximal in the set of all proper ideals of R, i.e. M is contained in exactly two ideals of R, namely M itself and the whole ring R. Every maximal ideal is in fact prime. In a principal ideal domain every nonzero prime ideal is maximal
The Jacobson radical of the ring Z/12Z is 6Z/12Z, which is the intersection of the maximal ideals 2Z/12Z and 3Z/12Z. Consider the ring C[t] ⊗ C C[x 1, x 2] x 1 2 +x 2 2 −1, where the second is the localization of C[x 1, x 2] by the prime ideal = (x 1 2 + x 2 2 − 1). Then, the Jacobson radical is trivial because the maximal ideals are ...
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Let A be a commutative Noetherian ring with unity. Then the following are equivalent. A is Artinian.; A is a finite product of commutative Artinian local rings. [5]A / nil(A) is a semisimple ring, where nil(A) is the nilradical of A.