Abstract Algebra MCQs

श्रेणी: Mathematics | विषय: Mathematics | टॉपिक: Abstract Algebra

3 प्रश्न

Mathematics • Abstract Algebra

Which one of the following statements is false? Which one of the following statements is false?

A. A field (F, +, .) always has a multiplicative identity. A field (F, +, .) always has a multiplicative identity. B. M_nxn(R), the space of all n x n real-valued matrices under matrix addition and multiplication, forms an integral domain. M_nxn(R), the space of all n x n real-valued matrices under matrix addition and multiplication, forms an integral domain. C. The set of complex numbers forms an algebraically closed field. The set of complex numbers forms an algebraically closed field. D. For a group (G, .), the identity element is always unique. For a group (G, .), the identity element is always unique.

सही उत्तर: B Correct Answer: B

For n greater than 1, the ring of n x n real matrices has zero divisors and is noncommutative, so it is not an integral domain.

Mathematics • Abstract Algebra

The set of units in Z_14, under the usual operations of addition and multiplication modulo 14, is: The set of units in Z_14, under the usual operations of addition and multiplication modulo 14, is:

A. Z_14 Z_14 B. {1} {1} C. {1, 13, 3, 5, 11, 9} {1, 13, 3, 5, 11, 9} D. {1, -1, 3, 5, 2, 7} {1, -1, 3, 5, 2, 7}

सही उत्तर: C Correct Answer: C

The units modulo 14 are exactly the residue classes relatively prime to 14: 1, 3, 5, 9, 11 and 13.

Mathematics • Abstract Algebra

Which one of the following statements is false? Which one of the following statements is false?

A. Intersection of any two subgroups is a subgroup. Intersection of any two subgroups is a subgroup. B. For any ring (R,+,.), the relation a(b-c)=0 implies b=c. For any ring (R,+,.), the relation a(b-c)=0 implies b=c. C. If (G,.) is abelian and H,K are subgroups of G, then {hk: h in H, k in K} is a subgroup of G. If (G,.) is abelian and H,K are subgroups of G, then {hk: h in H, k in K} is a subgroup of G. D. Z4 under addition modulo 4 is cyclic. Z4 under addition modulo 4 is cyclic.

सही उत्तर: B Correct Answer: B

Cancellation need not hold in an arbitrary ring because rings can have zero divisors. Thus a(b-c)=0 does not always imply b=c.

Abstract Algebra MCQs

Abstract Algebra MCQs

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