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Let R be a field of real numbers and M₂(R) be a ring of 2 x 2 matrices over R under usual operations, then: a) If S is a subring of M,(R). then S is an ideal of M₂(R). b) If S is an ideal of M₂(R), the S is a subring of M₂(R). c) There are no two non-trivial subrings S, and S2 of M₂(R) such that S, US, is a subring of R. dy For any two ideals I, and 12 of M₂(R). then I, U 12 is not ideal of M₂(R)

Let R be a field of real numbers and M₂(R) be a ring of 2 x 2 matrices over R under usual operations, then: a) If S is a subring of M,(R). then S is an ideal of M₂(R). b) If S is an ideal of M₂(R), the S is a subring of M₂(R). c) There are no two non-trivial subrings S, and S2 of M₂(R) such that S, US, is a subring of R. dy For any two ideals I, and 12 of M₂(R). then I, U 12 is not ideal of M₂(R)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let R be a field of real numbers and M₂(R) be a ring of 2 x 2 matrices over R under usual operations, then: a) If S is a subring of M,(R). then S is an ideal of M₂(R). b) If S is an ideal of M₂(R), the S is a subring of M₂(R). c) There are no two non-trivial subrings S, and S2 of M₂(R) such that S, US, is a subring of R. dy For any two ideals I, and 12 of M₂(R). then I, U 12 is not ideal of M₂(R) Let Z, is a ring of integers mod 5 and f(x)eZs(x) such that f(x) = x²+x+1, then: a) f(x) is a field. b) f(x) has only two roots. c) f(x) has more than two roots. d) No one of the above. 1. Let 7 be a discrete topology on a set X and A is non empty proper subset of X. then; a) Int(A) and b(A) = A b) Int(A) A and b(A)=A. c) Int(A) A and b(A)= d) Int(A) op and b(A) = op 2 10, 5 55 4.4
Transcribed Image Text:Let R be a field of real numbers and M₂(R) be a ring of 2 x 2 matrices over R under usual operations, then: a) If S is a subring of M,(R). then S is an ideal of M₂(R). b) If S is an ideal of M₂(R), the S is a subring of M₂(R). c) There are no two non-trivial subrings S, and S2 of M₂(R) such that S, US, is a subring of R. dy For any two ideals I, and 12 of M₂(R). then I, U 12 is not ideal of M₂(R) Let Z, is a ring of integers mod 5 and f(x)eZs(x) such that f(x) = x²+x+1, then: a) f(x) is a field. b) f(x) has only two roots. c) f(x) has more than two roots. d) No one of the above. 1. Let 7 be a discrete topology on a set X and A is non empty proper subset of X. then; a) Int(A) and b(A) = A b) Int(A) A and b(A)=A. c) Int(A) A and b(A)= d) Int(A) op and b(A) = op 2 10, 5 55 4.4
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