Question

Consider the following conditions on two proper non-zero ideals \( J_1 \) and \( J_2 \) of a non-zero commutative ring \( R \): P: For any \( r_1, r_2 \in R \), there exists a unique \( r \in R \) such that \( r - r_1 \in J_1 \) and \( r - r_2 \in J_2 \). Q: \( J_1 + J_2 = R \) Then, which of the following statements is TRUE?

• A. P implies Q but Q does not imply P
• B. Q implies P but P does not imply Q
• C. P implies Q and Q implies P
• D. P does not imply Q and Q does not imply P

Hint:

In ideal theory, adding two ideals can generate the entire ring, but the uniqueness and specific conditions for generating elements may not always follow.

Answer 1

The Correct Option is A

Step 1: Understanding P and Q.
- Statement P means that for any two elements \( r_1 \) and \( r_2 \) in \( R \), there is a unique element \( r \in R \) that satisfies the conditions of belonging to the ideals \( J_1 \) and \( J_2 \). - Statement Q says that the sum of the two ideals \( J_1 \) and \( J_2 \) is the entire ring \( R \). Step 2: Analyzing the relationship between P and Q.
- If P is true, it implies that the conditions on the ideals \( J_1 \) and \( J_2 \) are satisfied, which leads to \( J_1 + J_2 = R \), hence Q holds. - However, Q does not necessarily imply P. For example, even if \( J_1 + J_2 = R \), there may not be a unique \( r \in R \) satisfying the condition in P. Thus, the correct answer is (A) P implies Q but Q does not imply P.