Questions: Ideal Operations

Question 1

In Z, what is the ideal (6) ∩ (10)?

• A. (2)
• B. (30)
• C. (60)
• D. (16)

Answer: (B)

(6) ∩ (10) consists of integers divisible by both 6 and 10, which means divisible by lcm(6, 10) = 30. So the intersection is (30). In a PID, (a) ∩ (b) = (lcm(a, b)), while (a)(b) = (ab) and (a) + (b) = (gcd(a, b)).

Question 2

For ideals I, J in a commutative ring, IJ ⊆ I ∩ J always holds.

• T. True
• F. False

Answer: True

Every element of IJ is a finite sum of products ab with a in I and b in J. Each such ab lies in I (since I is an ideal and a is in I) and in J (since J is an ideal and b is in J). So ab is in I ∩ J, and hence every element of IJ is in I ∩ J. Equality holds when I + J = R (the ideals are comaximal), by the Chinese Remainder Theorem.

Question 3

What is the radical of the ideal (12) in Z?

Model answer: (6), since the primes containing (12) are (2) and (3), and √(12) = (2) ∩ (3) = (6).

The radical √(n) in Z is (rad(n)), where rad(n) is the product of distinct prime factors of n. Since 12 = 2^2 * 3, we get rad(12) = 6. Equivalently, √(12) = {m in Z : m^k in (12) for some k} = (6), since 6^2 = 36 is divisible by 12 but no smaller positive integer has this property.

Question 4

If P is a prime ideal and P ⊇ I ∩ J, then P ⊇ I or P ⊇ J.

• T. True
• F. False

Answer: True

Suppose P does not contain I and does not contain J. Then there exist a in I \ P and b in J \ P. But ab is in IJ ⊆ I ∩ J ⊆ P, and since P is prime, a in P or b in P -- contradiction. This 'prime avoidance' property is used constantly in arguments about prime ideals.

Question 5

Compute the ideal quotient (x^2, xy) : (x) in the polynomial ring k[x, y].

Model answer: (x, y), since (x^2, xy) : (x) = {f in k[x, y] : fx in (x^2, xy)} = {f : fx = gx^2 + hxy for some g, h} = (x, y).

We need all f such that fx lies in (x^2, xy). This means fx is a k[x,y]-linear combination of x^2 and xy, so f is a linear combination of x and y. Thus the quotient is (x, y). Ideal quotients are dual to products and play a key role in primary decomposition and computational algebra.