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?
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?
Show Hint
- \( P1 \Rightarrow P2 \)
- \( P2 \Rightarrow P3 \)
- \( P3 \Rightarrow P4 \)
- \( P4 \Rightarrow P1 \)
The Correct Option is A, B, C
Solution and Explanation
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
\[ \boxed{(A) \ P1 \Rightarrow P2}, \quad \boxed{(B) \ P2 \Rightarrow P3}, \quad \boxed{(C) \ P3 \Rightarrow P4} \]
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S2: For every compact subset \( K \) of \( \mathbb{R} \), \( g^{-1}(K) \) is compact. Then, which one of the following is correct?Consider the following subsets of the Euclidean space \( \mathbb{R}^4 \):
\( S = \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1^2 + x_2^2 + x_3^2 - x_4^2 = 0 \} \),
\( T = \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1^2 + x_2^2 + x_3^2 - x_4^2 = 1 \} \),
\( U = \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1^2 + x_2^2 + x_3^2 - x_4^2 = -1 \} \).
Then, which one of the following is TRUE?
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