Does Maximal Implies Prime?
Every Prime Ideal Is Maximal If An=A for Any Element a in the Commutative Ring Let R Be a Commutative Ring with Identity 1≠0. Suppose That for Each Element...
Every Prime Ideal is Maximal if an=a for any Element a in the Commutative Ring Let R be a commutative ring with identity 1≠0. Suppose that for each element a∈R, there exists an integer n>1 depending on a. Then prove that every prime ideal is a maximal ideal.
Is a maximal ideal a prime ideal?
As with commutative rings, maximal ideals are prime, and also prime ideals contain minimal prime ideals. A ring is a prime ring if and only if the zero ideal is a prime ideal, and moreover a ring is a domain if and only if the zero ideal is a completely prime ideal.
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Is 4Z a prime ideal?
The ideals Z,4Z,8Z ⊂ Z are neither prime nor maximal, so that the ideals Z/8Z,4Z/8Z,(0) ⊂ Z/8Z are neither prime nor maximal.
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