MCQ on ring theory
1. (Z,+,•) is ? (1 point)
◯ field
◯ Ring
◯ integral domain
◯ Ring and integral domain
2. A integral domain is field if________ (1 point)
◯ finite
◯ infinit
3. If in a ring with unit (xy)²=x²y² ∀x,y∈R,then, (1 point)
◯ R is commutative ring
◯ R is an integral domain
◯ R is a field
◯ Not of this
4. Which of following is not a ring? (1 point)
◯ (N,+,•)
◯ (Z,+,•)
◯ (Q,+,•)
◯ (R,+,•)
5. In a ring (R,+,•) which of following is correct? (1 point)
◯ (a+b)²⁼a²+ab+ba+b²
◯ (a+b)(a-b)=a²-ab+ba-b²
◯ Both (a) and (b) correct
◯ Both (a)&(b) are not correct
6. The characteristic of a ring (Z₃,+₃,•₃) is _______ (1 point)
◯ 0
◯ 1
◯ 2
◯ 3
7. Which of following is a finite field? (1 point)
◯ Z₇
◯ Z₈
◯ Z₉
◯ Z₁₀
8. The ring of Gaussian integral ( Z[i],+,•) where Z[i]={a+ib: a,b∈Z} (1 point)
◯ a field not integral domain
◯ an integral domain not a field
◯ Both field and integral domain
◯ Neither field nor integral domain
9. The intersection of two ideals is_______ (1 point)
◯ An ideal
◯ not an ideal
◯ A left ideal but not right
◯ Right ideal not left
10. U is two sides ideal of ring R if________ (1 point)
◯ (U,+) is subgroup
◯ For r∈ R, u∈U ru∈U
◯ For r∈ R, u∈U ur∈U
◯ A,B and C
11. Let ∅: R--->R' be ring homomorphisThen kernel of I(∅) is (1 point)
◯ Subgroup of R
◯ Ideal of R
◯ Both A and B
◯ Non of them
12. U be an ideal of ring R,then R/U is (1 point)
◯ Ideal of R
◯ Ring
◯ Both
◯ None of them
13. R be a integral domain and U be an ideal of R then R/U is (1 point)
◯ integral domain
◯ need not be integral domain
14. U be a ideal of R and let 1 be a unite element in R such that 1∈U (1 point)
◯ U=R
◯ U⊂R
◯ R⊂U
◯ A,B and C all are correct
15. Minimum number of elements in a field. (1 point)
◯ 0
◯ 1
◯ 2
◯ None of them
16. Minimum number of elements in a ring. (1 point)
◯ 0
◯ 1
◯ 2
◯ None of them
17. For any integer Zₙ is a field if n is _____ (1 point)
◯ An even number
◯ An odd number
◯ A prime number
◯ A perfect number
18. The number of unit element of ring Z is ______ (1 point)
◯ 1
◯ 2
◯ finite but more than 2
◯ Infinit
19. Which of following is correct? (1 point)
◯ Every integral domain is field
◯ Every ring is an integral domain
◯ Every ring is a field
◯ Every field is an integral domain
20. The characteristic of an integral domain is_____ (1 point)
◯ Either 0 or prime number
◯ Either 0 or positive number
◯ Either 1 or a prime number
◯ Either 1 or a positive number
21. A commutative ring with unit element and without zero divisors is called (1 point)
◯ Ring
◯ Field
◯ Integral domain
◯ Sub field
22. The singleton {0} is a ring with respect to _____ (1 point)
◯ Addition
◯ Multiplication
◯ Both A and B
◯ None
23. M₂(R) is an integral domain (1 point)
◯ True
◯ False
24. The set of units is J(i) is ______ (1 point)
◯ {+1,-1,+i,-i}
◯ {-1,+1}
◯ {+i,-i}
25. The polynomical f(x)=a₀+a₁x+......aₙxⁿ,where the a₁,a₂,a₃,...,aₙ are integers is said
to be primitive if the g.c.d. of a₁,a₂,...,aₙ is
(1 point)
◯ 1
◯ 2
◯ 3
◯ 4
26. Polynomial x²+1 is irreducible over the real field. (1 point)
◯ True
◯ False
27. Polynomial x²+1 is irreducible over the complex field. (1 point)
◯ True
◯ False
28. Polynomial x²+1 is irreducible over Z₇ (1 point)
◯ True
◯ False
29. Polynomial x²+x+1 is irreducible over z₂ (1 point)
◯ True
◯ False
30. The number of unit element of ring Z is ______ (1 point)
◯ 2
◯ 3
◯ 0
◯ None of them
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