2 let (Z+) is ring show that every ideal is cyclic or not? A Sield K has only the trivial ideals and K? Z contains no minimal ideal. Qz has no Minimal and no Maximal submodules fin the Minimal and Maximal ideal of Zn, S.T n = {3, 4, 14, 20} 18 to

Exercises Find two ideals and of the ring such that is not an ideal of . is an ideal of .

. a. Let, and . Show that and are only ideals of and hence is a maximal ideal. b. Show that is not a field. Hence Theorem is not true if the condition that is commutative is removed. Theorem 6.22 Quotient Rings That are Fields. Let be a commutative ring with unity, and let be an ideal of . Then is a field if and only if is a maximal ideal of .

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 ] }.

18. Find subrings and of such that is not a subring of .

29. Let be the set of Gaussian integers . Let . a. Prove or disprove that is a substring of . b. Prove or disprove that is an ideal of .

Prove that every ideal of n is a principal ideal. (Hint: See corollary 3.27.)

Find all maximal ideals of .

Let I1 and I2 be ideals of the ring R. Prove that the set I1I2=a1b1+a1b2+...+anbnaiI1,biI2,nZ+ is an ideal of R. The ideal I1I2 is called the product of ideals I1 and I2.

Let R be a commutative ring with unity whose only ideals are {0} and R Prove that R is a field.(Hint: See Exercise 30.)