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Let the set Z/ nZ denote the ring of integers modulo n under addition and multiplication modulo n then, Z/ 9Z is not a subring of Z/12Z because
  • a)
    Z/9Z is not a subset of Z/12Z
  • b)
    gcd(9, 12) = 3 ≠ 1
  • c)
    12 is not a power of 3
  • d)
    9 does not divide 12
Correct answer is option 'D'. Can you explain this answer?
Most Upvoted Answer
Let the set Z/ nZdenote the ring of integers modulo nunder addition an...
A) Z/9Z is a subset of Z/12Z. Both sets consist of residue classes modulo 12, and Z/9Z is a subset of Z/12Z because every element in Z/9Z is also an element in Z/12Z.

b) The fact that gcd(9, 12) = 3 is not a sufficient reason to conclude that Z/9Z is not a subring of Z/12Z. The gcd(9, 12) = 3 only indicates that 3 is a common divisor of 9 and 12, but it does not directly relate to the subring property.

The correct reason why Z/9Z is not a subring of Z/12Z is that the multiplication operation in Z/9Z does not satisfy the ring axioms when considered as a subset of Z/12Z. Specifically, there exist elements in Z/9Z whose products do not belong to Z/9Z, violating the closure property of a subring.
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Let the set Z/ nZdenote the ring of integers modulo nunder addition and multiplication modulo n then, Z/ 9Z is not a subring of Z/12Z becausea)Z/9Zis not a subset of Z/12Zb)gcd(9, 12) = 3 ≠ 1c)12 is not a power of 3d)9 does not divide 12Correct answer is option 'D'. Can you explain this answer?
Question Description
Let the set Z/ nZdenote the ring of integers modulo nunder addition and multiplication modulo n then, Z/ 9Z is not a subring of Z/12Z becausea)Z/9Zis not a subset of Z/12Zb)gcd(9, 12) = 3 ≠ 1c)12 is not a power of 3d)9 does not divide 12Correct answer is option 'D'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let the set Z/ nZdenote the ring of integers modulo nunder addition and multiplication modulo n then, Z/ 9Z is not a subring of Z/12Z becausea)Z/9Zis not a subset of Z/12Zb)gcd(9, 12) = 3 ≠ 1c)12 is not a power of 3d)9 does not divide 12Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let the set Z/ nZdenote the ring of integers modulo nunder addition and multiplication modulo n then, Z/ 9Z is not a subring of Z/12Z becausea)Z/9Zis not a subset of Z/12Zb)gcd(9, 12) = 3 ≠ 1c)12 is not a power of 3d)9 does not divide 12Correct answer is option 'D'. Can you explain this answer?.
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