Let R b e the ring of all 2 × 2 matri es with integer entries and letS ={(0 ab 0) ∣∣∣∣∣ a, b ∈ Z}b e a subset of R. Determine whether or not S is a subring of R, justifyingyour answer.

To determine whether S is a subring of R, we need to check three properties:

1. The zero element of R is in S.

2. S is closed under subtraction.

3. S is closed under multiplication.

4. The zero element of R is the 2x2 matrix with all entries equal to zero. This matrix is in S because we can choose a = b = 0.

5. To check if S is closed under subtraction, consider two arbitrary elements of S, (0 a1 b1 0) and (0 a2 b2 0). Their difference is (0 a1 b1 0) - (0 a2 b2 0) = (0 a1-a2 b1-b2 0), which is in S because a1 - a2 and b1 - b2 are integers.

6. To check if S is closed under multiplication, consider two arbitrary elements of S, (0 a1 b1 0) and (0 a2 b2 0). Their product is (0 a1 b1 0) * (0 a2 b2 0) = (0 0 0 0), which is in S.

Therefore, S is a subring of R.