IDEAL practice concepts

Question 1

Find a maximal ideal of Z × Z.

Answer 1

Zx2Z is a maximal ideal of ZxZ for the factor ring is isomorphic to Z2 which is a field

Question 2

Find a prime ideal of Z × Z that is not maximal.

Answer 2

Zx{0} is a prime ideal of ZxZ that is not maximal for the factor ring is isomorphic to Z which is an integral domain, but not a field

Question 3

Find a nontrivial proper ideal of Z × Z that is not prime.

Answer 3

Zx4Z is a proper ideal of ZxZ that is not a prime for the factor ring is isomorphic to Z4 which has divisors of zero

Question 4

Is Q[x]/⟨x^2 − 5x + 6⟩ a field? Why?

Answer 4

It’s not a field because x^2-5x +6=(x-2)(x-3) is not an irreducible polynomial so the ideal (x^2-5x+6) is not maximal

Question 5

Is Q[x]/⟨x^2 − 6x + 6⟩ a field? Why?

Answer 5

It is a field because the polynomial x^2-6x+6 is irreducible so it’s is a maximal ideal

Question 6

Prove that if R is a commutative ring with unity and a ∈ R, then ⟨a⟩ = {ra|r ∈ R} is an ideal in R.

Answer 6

Show <a> is a subgroup of <R,+>,</a>

ar+as=a(r+s), show <a> is closed under add.
0=0.a∈<a>.-(ra)=(-r)a implies that if an element is in <a>, then its additive inverse is also in <a>. So <a> is an additive subgroup of <R,+>. Now let ra∈<a> a s∈R. Then (ra)s=s(ra)=(sr)a∈<a>. so <a> is an ideal</a></a></a></a></a></a></a></a>

Question 7

Let R be a finite commutative ring with unity. Show that every prime ideal in R is a maximal ideal.

Answer 7

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Let N be a prime ideal in a finite commutative ring R with unity. Then R/N is a finite integral domain, and therefore a field, and therefore N is a maximal ideal.