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 3 Let D be a principal ideal domain. Prove that every non-zero prime ideal in D is a maximal ideal in D. 3 Let $R = \mathbb{Z}[x]$ and let $I = \langle x \rangle$ be the ideal in $R$ generated by $x$. Show $I$ is a prime ideal but not a maximal. 2 Let D be a principal ideal domain and let p be in D. Prove p is a prime element if and only if p is an irreducible element. 2 Suppose $R$ is a commutative ring with unity such that $a^2=a$ for $a \in R$. Prove if $I$ is a prime ideal in $R$ then $R/I$ has 2 elements 1 Newtons's Method

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 +15 Let D be a principal ideal domain. Prove that every non-zero prime ideal in D is a maximal ideal in D. +5 Let R be a commutative ring with unity, Prove {0} is a prime ideal in R if and only if R is an integral domain -2 Prove R/I is commutative +5 Let D be a principal ideal domain and let p be in D. Prove p is a prime element if and only if p is an irreducible element.

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