left ideal but not right ideal. right ideal but not left ideal. both left and…
2026
left ideal but not right ideal.
right ideal but not left ideal.
both left and right ideal.
neither left ideal nor right ideal.
- A.
1
- B.
2
- C.
3
- D.
4
Attempted by 29 students.
Show answer & explanation
Correct answer: D
The correct answer is D (statement 4): neither left ideal nor right ideal.
The set S contains all diagonal 2 × 2 integer matrices, so S is a subring of M₂(ℤ). However, to be a left ideal, for every matrix M ∈ M₂(ℤ) and every A ∈ S, the product MA must again be in S. To be a right ideal, AM must again be in S for every such M and A.
For the left-ideal condition, take A = [[1, 0], [0, 0]] ∈ S and M = [[0, 0], [1, 0]] ∈ M₂(ℤ). Then MA = [[0, 0], [1, 0]], which is not diagonal, so MA ∉ S. Hence S is not a left ideal.
For the right-ideal condition, take A = [[1, 0], [0, 0]] ∈ S and N = [[0, 1], [0, 0]] ∈ M₂(ℤ). Then AN = [[0, 1], [0, 0]], which is not diagonal, so AN ∉ S. Hence S is not a right ideal.
Therefore, S is neither a left ideal nor a right ideal.