Prove that every non-zero prime ideal of Z is a maximal ideal.
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- True or false Label each of the following statements as either true or false. 6. Every ideal of is a principal ideal.27. If is a commutative ring with unity, prove that any maximal ideal of is also a prime ideal.Factor each of the polynomial in Exercise as a product of its leading coefficient and a finite number of monic irreducible polynomial over the field of rational numbers.
- Prove that [ x ]={ a0+a1x+...+anxna0=2kfork }, the set of all polynomials in [ x ] with even constant term, is an ideal of [ x ]. Show that [ x ] is not a principal ideal; that is, show that there is no f(x)[ x ] such that [ x ]=(f(x))={ f(x)g(x)g(x)[ x ] }. Show that [ x ] is an ideal generated by two elements in [ x ] that is, [ x ]=(x,2)={ xf(x)+2g(x)f(x),g(x)[ x ] }.If R is a finite commutative ring with unity, prove that every prime ideal of R is a maximal ideal of R.Prove that if R is a field, then R has no nontrivial ideals.