Final answer:
The subset {0, 8, 16} is a subring of ℤ24 with exactly three elements. This set is closed under addition and multiplication modulo 24 and contains the additive identity 0, but it does not contain a multiplicative identity.
Step-by-step explanation:
The question involves finding a subring with exactly 3 elements in the ring ℤ24 (the integers modulo 24). In ring theory, a subring is a subset that is itself a ring, closed under addition and multiplication, contains the additive identity, and also contains the additive inverses of its elements. To find such a subring, note that any subring of ℤ24 must itself be a set of equivalence classes modulo 24.
One possible subring of ℤ24 with three elements is {0, 8, 16}. We can see that this set is closed under addition and multiplication modulo 24:
- 8 + 8 = 16 mod 24
- 8 + 16 = 24 ≈ 0 mod 24
- 16 + 16 = 32 ≈ 8 mod 24
- 8 × 8 = 64 ≈ 16 mod 24
- 8 × 16 = 128 ≈ 8 mod 24
- 16 × 16 = 256 ≈ 16 mod 24
It also contains 0, which is the additive identity in ℤ24. However, this subring does not contain a multiplicative identity, which would be an element 'e' such that for any element 'a' in the subring, e × a = a. The only element that satisfies this in ℤ24 is 1, and since 1 is not in our set, this subring does not have a multiplicative identity.