Question

All rings considered below are assumed to be associative and commutative with \( 1 \neq 0 \). Further, all ring homomorphisms map 1 to 1.

Consider the following statements about such a ring \( R \):

P1: \( R \) is isomorphic to the product of two rings \( R_1 \) and \( R_2 \).
P2: \( \exists r_1, r_2 \in R \) such that \( r_1^2 = r_1 \neq 0 \), \( r_2^2 = 0 \), and \( r_1 + r_2 = 1 \).
P3: \( R \) has ideals \( I_1, I_2 \subset R \) with \( R \neq I_1 \), \( (0) \neq I_2 \), and \( R = I_1 + I_2 \) and \( I_1 \cap I_2 = (0) \).
P4: \( \exists a, b \in R \) with \( a \neq 0 \), \( b \neq 0 \) such that \( ab = 0 \).

Then, which of the following is/are TRUE?

• A. \( P1 \Rightarrow P2 \)
• B. \( P2 \Rightarrow P3 \)
• C. \( P3 \Rightarrow P4 \)
• D. \( P4 \Rightarrow P1 \)

Hint:

When working with ring homomorphisms and ideals, it is important to check the properties of the ring and its decomposition before drawing conclusions about its structure.

Answer 1

The Correct Option is A, B, C

Step 1: \( P1 \Rightarrow P2 \)
If \( R \) is isomorphic to the product of two rings \( R_1 \) and \( R_2 \), then \( R \cong R_1 \times R_2 \) where we can take \( r_1 = (1, 0) \) and \( r_2 = (0, 1) \), fulfilling the conditions of \( P2 \). Therefore, (A) is TRUE.

Step 2: \( P2 \Rightarrow P3 \)
The condition in \( P2 \) gives us the existence of elements \( r_1 \) and \( r_2 \) with the required properties. This implies that \( R \) is a direct sum of ideals. Therefore, (B) is TRUE.

Step 3: \( P3 \Rightarrow P4 \)
The structure given in \( P3 \) implies the existence of idempotent elements, which satisfy the conditions for \( P4 \) (where \( ab = 0 \)). Therefore, (C) is TRUE.

Step 4: \( P4 \Rightarrow P1 \)
The condition in \( P4 \) does not necessarily imply that \( R \) is isomorphic to the product of two rings. Hence, (D) is FALSE.

Final Answer
\[ \boxed{(A) \ P1 \Rightarrow P2}, \quad \boxed{(B) \ P2 \Rightarrow P3}, \quad \boxed{(C) \ P3 \Rightarrow P4} \]