Let R be a commutative ring with identity. Let S be a multiplicatively closedset such that 0 $$\notin$$​ S. Let I be an ideal which is maximal with respect to thecondition thatS ∩ I = ∅.

Which of the following is necessarily true?

Correct option is B

Let S ∩ I = ∅, where I is maximal with respect to this condition.

Case 1: If R = Z 

let I = ⟨2⟩ = {2n : n ∈ Z} and S = {5n : n ∈ Z}.

Here, S ∩ I = ∅, but I = ⟨1⟩ = Z, which contradicts the condition.

Thus, (C)is discarded.

Case 2: Consider I ⊆ $$I_1$$​ such that S ∩ $$I_1\neq$$ ∅. Take R = Z

I = ⟨2⟩, andS = {5n : n ∈ Z}. Assume a ∈ Z:

• If a is odd, then a ∈ $$I_1$$​.

• If a + 1 is even, a + 1 ∈ $$I_1$$​.

• Then, a + 1 − a = 1 ∈ $$I_1$$​.

Thus, $$I_1$$​ = R. However, I = ⟨0⟩ contradicts (D).

Hence, (D) is discarded.

Now, in the context of a commutative ring with unity (CRU), every maximalideal is a prime ideal.

Thus, (a) is also discarded.

​$$\textbf{Final Answer:}$$​ Option (B) is correct .