Examples of Rings: Delve into the Fundamentals of Algebraic Structures

Question 1

Identify the set that does not constitute a ring

Accepted Answer

Finite groups under composition

Answer

Integers under addition and multiplication

Answer

Rational numbers under addition and multiplication

Answer

Complex numbers under addition and multiplication

Question 2

Determine the additive inverse of 5 in the ring of integers

Accepted Answer

-5

Answer

5

Answer

0

Answer

1

Question 3

What is the characteristic of the ring of rational numbers?

Accepted Answer

0

Answer

1

Answer

2

Answer

3

Question 4

In the ring of polynomials over a field, which ideal is generated by the polynomial x^2 - 1?

Accepted Answer

(x^2 - 1)

Answer

(x + 1)

Answer

(x - 1)

Answer

(x - 1, x + 1)

Question 5

Find the homomorphism between the rings of integers and rational numbers

Accepted Answer

f(x) = x/1

Answer

f(x) = x

Answer

f(x) = x^2

Answer

f(x) = 1/x

Question 6

Which of the following is an example of a ring, where addition and multiplication are both defined and follow the ring properties?

Accepted Answer

The set of complex numbers under addition and multiplication

Answer

The set of integers under addition only

Answer

The set of rational numbers under subtraction only

Answer

The set of real numbers under multiplication only

Question 7

Which of these sets, with standard addition and multiplication, does NOT form a ring?

Accepted Answer

Natural numbers

Answer

Integers

Answer

Rational numbers

Answer

Complex numbers

Question 8

Which of the following is NOT an example of a ring?

Accepted Answer

The set of 2x2 matrices with real entries under addition and matrix multiplication

Answer

The set of integers under addition and multiplication

Answer

The set of rational numbers under addition and multiplication

Question 9

Which of the following is NOT a ring?

Accepted Answer

The set of real numbers under addition and multiplication

Answer

The set of integers under addition and multiplication

Answer

The set of complex numbers under addition and multiplication

Answer

The set of rational numbers under addition and multiplication

Question 10

Which of the following is an example of a finite ring?

Accepted Answer

The ring of 2x2 matrices with real entries of the form [a b; 0 d]

Answer

The ring of integers under addition and multiplication

Answer

The ring of rational numbers under addition and multiplication

Answer

The ring of complex numbers under addition and multiplication

Question 11

Which of the following rings is a subring of the ring of integers?

Accepted Answer

The ring of even integers under addition and multiplication

Answer

The ring of rational numbers under addition and multiplication

Answer

The ring of complex numbers under addition and multiplication

Answer

The ring of 2x2 matrices with real entries under addition and multiplication

Question 12

Which of the following rings is an ideal of the ring of integers?

Accepted Answer

The ring of even integers under addition and multiplication

Answer

The ring of complex numbers under addition and multiplication

Answer

The ring of 2x2 matrices with real entries under addition and multiplication

Answer

The ring of rational numbers under addition and multiplication

Question 13

Which of the following sets with the given operations does NOT form a ring?

Accepted Answer

The set of 2 × 2 matrices over the real numbers under addition and matrix multiplication

Answer

The set of integers under addition and multiplication

Answer

The set of rational numbers under addition and multiplication

Answer

The set of complex numbers under addition and multiplication

Question 14

Which set does not satisfy the definition of a ring?

Accepted Answer

Odd integers under addition and multiplication

Answer

Integers under addition and multiplication

Answer

Rational numbers under addition and multiplication

Question 15

Which of the following is NOT an example of a ring in mathematics?

Accepted Answer

Set of natural numbers with standard addition and multiplication

Answer

Set of integers with standard addition and multiplication

Answer

Set of rational numbers with standard addition and multiplication

Answer

Set of complex numbers with standard addition and multiplication

Question 16

Consider the set Z of integers under addition and multiplication operations. Which property holds true for Z?

Accepted Answer

Z is a commutative ring with unity.

Answer

Z is not a ring.

Answer

Z is a non-commutative ring.

Answer

Z is a field.

Question 17

Let R represent the set of all 2x2 matrices with real entries. Which property is NOT applicable to R under matrix addition and multiplication?

Accepted Answer

R is a field.

Answer

R has a multiplicative identity.

Answer

R is commutative under addition.

Answer

R is a ring.

Question 18

For the set C of complex numbers, which operations make C a ring?

Accepted Answer

Addition and multiplication

Answer

Subtraction and division

Answer

Multiplication and division

Answer

Addition and subtraction

Question 19

Identify an example of a subring of the ring of integers:

Accepted Answer

Set of even integers

Answer

Set of odd integers

Answer

Set of rational numbers

Answer

Set of prime numbers

Question 20

Let R be a ring with unity. Which statement is equivalent to R being a field?

Accepted Answer

All non-zero elements of R have multiplicative inverses.

Answer

R is commutative under addition.

Answer

R has no zero divisors.

Question 21

Identify the property that does NOT apply to the zero element in a ring:

Accepted Answer

It has a multiplicative inverse.

Answer

It is the multiplicative identity.

Answer

It is unique.

Answer

It is the additive identity.

Question 22

Which of the following is NOT an example of a ring?

Accepted Answer

The set of polynomials with real coefficients is not an example of a ring because it does not have a multiplicative inverse for every element.

Answer

The set of complex numbers with addition and multiplication is an example of a ring.

Answer

The set of integers with addition and multiplication is an example of a ring.

Answer

The set of rational numbers with addition and multiplication is an example of a ring.

Question 23

Which of the following rings is a field?

Accepted Answer

The set of rational numbers is a field because it satisfies all the properties of a field, including commutativity, associativity, distributivity, and the existence of multiplicative inverses.

Answer

The set of integers is not a field because it does not have multiplicative inverses for all elements.

Answer

The set of polynomials is not a field because it does not have multiplicative inverses for all elements.

Answer

The set of quaternions is not a field because it is non-commutative.

Question 24

Which of the following is an example of a non-associative ring?

Accepted Answer

The set of matrices is a non-associative ring because for some matrices a, b, and c, (a * b) * c does not equal a * (b * c).

Answer

The set of integers is an associative ring.

Answer

The set of rational numbers is an associative ring.

Answer

The set of complex numbers is an associative ring.

Question 25

Which of the following rings is a division ring?

Accepted Answer

The set of rational numbers is a division ring because every non-zero element has a multiplicative inverse.

Answer

The set of quaternions is not a division ring because it is non-commutative.

Answer

The set of integers is not a division ring because not all elements have multiplicative inverses.

Answer

The set of polynomials is not a division ring because not all elements have multiplicative inverses.

Question 26

Which of the following rings is a principal ideal domain?

Accepted Answer

The set of integers is a principal ideal domain because every ideal is generated by a single element.

Answer

The set of rational numbers is not a principal ideal domain.

Answer

The set of polynomials is not a principal ideal domain.

Answer

The set of quaternions is not a principal ideal domain.

Question 27

Which of the following rings is a Euclidean domain?

Accepted Answer

The set of integers is a Euclidean domain because there is a norm function that can measure the size of elements and a division algorithm for any two elements.

Answer

The set of polynomials is not a Euclidean domain.

Answer

The set of rational numbers is not a Euclidean domain.

Question 28

Which of the following rings is a unique factorization domain?

Accepted Answer

The set of integers is a unique factorization domain because every element can be uniquely factored into a product of prime elements.

Answer

The set of rational numbers is not a unique factorization domain.

Answer

The set of quaternions is not a unique factorization domain.

Answer

The set of polynomials is not a unique factorization domain.

Question 29

Which of the following is an example of a ring with unity?

Accepted Answer

Set of complex numbers under addition and multiplication

Answer

Set of rational numbers without 1 under addition and multiplication

Answer

Set of natural numbers excluding 1 under addition

Answer

Set of integers without 0 under addition and multiplication

Question 30

Which of the following statements about the ring of rational numbers is correct?

Accepted Answer

It is a field.

Answer

It is a principal ideal ring.

Answer

It is a Dedekind ring.

Answer

It is a Euclidean ring.

Question 31

Which of the following is an example of a finite ring?

Accepted Answer

Set of integers modulo n under addition and multiplication

Answer

Set of polynomials over a field under addition and multiplication

Answer

Set of matrices over a field under addition and multiplication

Answer

Set of quaternions under addition and multiplication

Question 32

Which of the following is an example of a ring?

Accepted Answer

The set of integers

Answer

The set of vector spaces

Answer

The set of irrational numbers

Answer

The set of mappings

Question 33

Which of the following is a subring of the ring of integers?

Accepted Answer

The set of even integers

Answer

The set of prime integers

Answer

The set of odd integers

Question 34

Which of the following is a commutative ring?

Accepted Answer

The set of integers

Answer

The set of vectors

Answer

The set of matrices

Answer

The set of functions

Question 35

Which of the following is a non-commutative ring?

Accepted Answer

The set of matrices

Answer

The set of rational numbers

Answer

The set of integers

Answer

The set of complex numbers

Question 36

Which of the following is an example of a polynomial ring?

Accepted Answer

The set of polynomials with real coefficients

Answer

The set of rational numbers

Answer

The set of complex numbers

Answer

The set of integers

Question 37

Which of the following is an example of a group ring?

Accepted Answer

The set of matrices with integer coefficients

Answer

The set of integers

Answer

The set of complex numbers

Answer

The set of rational numbers

Question 38

Which of the following is an example of a skew field?

Accepted Answer

The set of complex numbers

Answer

The set of rational numbers

Answer

The set of matrices

Answer

The set of integers

Question 39

Which of the following sets is a ring under the usual operations of addition and multiplication?

Accepted Answer

The set of integers (ℤ) with the operations of addition (+) and multiplication (×).

Answer

The set of natural numbers (ℕ) with the operations of addition (+) and multiplication (×).

Answer

The set of real numbers (ℝ) with the operations of addition (+) and multiplication (×).

Answer

None of the options provided.

Question 40

Let R be a ring with a multiplicative identity. Which of the following must be true for every element a in R?

Accepted Answer

There exists an element b in R such that a + b = 0. (This element b is called the additive inverse of a.)

Answer

There exists an element b in R such that a + b = 1. (This element b is called the additive identity of R.)

Answer

There exists an element b in R such that a × b = 0. (This element b is called the multiplicative inverse of a.)

Question 41

Let R be a ring with zero divisors. Which of the following statements is true?

Accepted Answer

R is not a division ring.

Answer

R has no zero divisors.

Answer

R is a field.

Answer

Every element in R is a unit.

Question 42

Which of the following is an example of a finite ring?

Accepted Answer

The ring ℤ/6ℤ, where ℤ/6ℤ is the set of equivalence classes [0], [1], [2], [3], [4], [5] under the operation of modular addition modulo 6.

Answer

The set of real numbers (ℝ) under the usual operations of addition (+) and multiplication (×).

Answer

The set of integers (ℤ) under the usual operations of addition (+) and multiplication (×).

Answer

The set of rational numbers (ℚ) under the usual operations of addition (+) and multiplication (×).

Question 43

Which of the following is a polynomial ring over a field F?

Accepted Answer

The ring F[x], which consists of polynomials with coefficients from the field F.

Answer

The ring ℝ[x], which consists of polynomials with real coefficients.

Answer

The ring ℤ[x], which consists of polynomials with integer coefficients.

Answer

The ring ℚ[x], which consists of polynomials with rational coefficients.

Question 44

Let R be a ring and let a, b ∈ R. If ab = ba, then which of the following must be true?

Accepted Answer

R is a commutative ring.

Answer

R is a Boolean ring.

Answer

R is a division ring.

Answer

R is a finite ring.

Question 45

Which of the following sets does NOT form a ring under the usual operations of addition and multiplication?

Accepted Answer

The set of real numbers with 0 excluded (ℝ{0}) under the usual operations of addition (+) and multiplication (×).

Answer

The set of integers (ℤ) under the usual operations of addition (+) and multiplication (×).

Answer

The set of rational numbers (ℚ) under the usual operations of addition (+) and multiplication (×).

Answer

The set of complex numbers (ℂ) under the usual operations of addition (+) and multiplication (×).

Question 46

Let R be a ring. Which of the following operations on R always produces a ring?

Accepted Answer

Taking the subring generated by a subset of R.

Answer

Taking the Cartesian product of R with itself.

Answer

Taking the union of R with another ring.

Question 47

Which of the following is an example of a commutative ring?

Accepted Answer

The set of integers (with usual addition and multiplication)

Answer

The set of complex numbers (with usual addition and multiplication)

Answer

The set of rational numbers (with usual addition and multiplication)

Question 48

Which of the following is not a property of a ring?

Accepted Answer

Commutativity of multiplication

Answer

Associativity of addition

Answer

Existence of an additive identity

Answer

Distributivity of multiplication over addition

Question 49

Which of the following is a subring of the ring of real numbers?

Accepted Answer

The set of rational numbers

Answer

The set of polynomials with real coefficients

Answer

The set of complex numbers

Answer

The set of integers

Question 50

Which of the following is an example of a ring without unity?

Accepted Answer

The set of 2x2 matrices with real entries (with usual addition and multiplication)

Answer

The set of integers (with usual addition and multiplication)

Answer

The set of rational numbers (with usual addition and multiplication)

Question 51

Which of the following is an example of a division ring?

Accepted Answer

The set of rational numbers (with usual addition and multiplication)

Answer

The set of complex numbers (with usual addition and multiplication)

Answer

The set of integers (with usual addition and multiplication)

Question 52

Which of the following is not a characteristic of a field?

Accepted Answer

Non-zero elements have multiplicative inverses

Answer

Distributivity of multiplication over addition

Answer

Commutativity of addition and multiplication

Answer

Associativity of addition and multiplication

Question 53

Which of the following is an example of a finite field?

Accepted Answer

The set of integers modulo 5 (with usual addition and multiplication modulo 5)

Answer

The set of rational numbers (with usual addition and multiplication)

Answer

The set of real numbers (with usual addition and multiplication)

Question 54

Which of the following is isomorphic to the ring of integers modulo 6?

Accepted Answer

The set of equivalence classes of integers modulo 6 under the relation a ≡ b (mod 6)

Answer

The set of real numbers (with usual addition and multiplication)

Answer

The set of rational numbers (with usual addition and multiplication)

Question 55

Which of the following is an example of a polynomial ring?

Accepted Answer

The set of polynomials with real coefficients (with usual addition and multiplication)

Answer

The set of integers (with usual addition and multiplication)

Answer

The set of rational numbers (with usual addition and multiplication)

Question 56

Which of the following is an example of a group ring?

Accepted Answer

The set of 2x2 matrices with real entries (with usual addition and matrix multiplication)

Answer

The set of rational numbers (with usual addition and multiplication)

Answer

The set of integers (with usual addition and multiplication)