Rings: Definition and Examples - Interactive Learning Quizzes

Question 1

In the ring of integers, what element serves as the multiplicative identity?

Accepted Answer

1

Answer

0

Answer

-1

Answer

2

Question 2

In the ring of polynomials over a field, what is the degree of the polynomial (x^2 - 1) * (x + 1)?

Accepted Answer

4

Answer

1

Answer

2

Answer

3

Question 3

What is the order of the ring (Z/6Z, +, *)?

Accepted Answer

6

Answer

2

Answer

4

Answer

12

Question 4

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

Accepted Answer

Even integers

Answer

Odd integers

Answer

Prime numbers

Answer

Complex numbers

Question 5

Identify the condition that is both necessary and sufficient for a commutative ring with unity to be considered a field.

Accepted Answer

All nonzero elements in the ring possess multiplicative inverses.

Answer

The ring is devoid of nonzero zero divisors.

Answer

The ring has a finite number of elements.

Answer

The ring exhibits associativity in its operations.

Question 6

Consider the ring of integers. Which property is not applicable to this ring?

Accepted Answer

Existence of multiplicative inverses for all non-zero elements

Answer

Commutative addition

Answer

Associative multiplication

Answer

Existence of a multiplicative identity (1)

Question 7

In the ring of polynomials over a field, which operation is not closed?

Accepted Answer

Exponentiation

Answer

Addition

Answer

Subtraction

Answer

Multiplication

Question 8

In the ring Z[x], which of the following polynomials is a unit?

Accepted Answer

x

Answer

x^2 + 1

Answer

2x + 1

Answer

x^2 - 1

Question 9

Which of the following is an ideal of the ring R = {a + bi | a, b ∈ Z}?

Accepted Answer

{a + bi | a, b ∈ 2Z}

Answer

{a + bi | a ∈ Z, b ∈ 2Z}

Answer

{a + bi | a, b ∈ 3Z}

Answer

{a + bi | a ∈ 2Z, b ∈ 3Z}

Question 10

Which property is NOT a defining characteristic of rings?

Accepted Answer

Commutativity of multiplication

Answer

Associativity of addition

Answer

Existence of an additive identity

Answer

Distributivity of multiplication over addition

Question 11

How does the Chinese Remainder Theorem help solve a system of linear congruences like this:
x ≡ a1 (mod m1)
x ≡ a2 (mod m2)
where gcd(m1, m2) = 1?

Accepted Answer

Find the multiplicative inverses of m1 and m2 modulo each other, then combine the congruences using a weighted average.

Answer

Multiply m1 and m2 to get m, then find x such that x ≡ a (mod m).

Answer

Calculate the least common multiple (LCM) of m1 and m2, then find x such that x ≡ a (mod lcm(m1, m2)).

Question 12

Which property is a defining characteristic of isomorphic rings?

Accepted Answer

Identical ring operations

Answer

Equal number of elements

Answer

Shared characteristic

Answer

Common additive identity

Question 13

Which of the following properties is NOT characteristic of a ring?

Accepted Answer

Commutativity of multiplication

Answer

Existence of an additive identity element

Answer

Existence of an additive inverse for each element

Answer

Associativity of addition

Question 14

In the ring R = {a + bi | a, b ∈ ℤ, i² = -1}, what is the additive identity?

Accepted Answer

0

Answer

1

Answer

i

Answer

1 + i

Question 15

In a ring with unity, which of the following properties holds for the multiplicative identity?

Accepted Answer

It is unique.

Answer

It may not exist in general.

Answer

There may be multiple multiplicative identities in the ring.

Answer

It is not necessarily equal to the additive identity of the ring.

Question 16

Identify an example of a non-commutative ring.

Accepted Answer

The ring of 2 × 2 matrices over the real numbers

Answer

The ring of integers

Answer

The ring of polynomials with real coefficients

Answer

The ring of rational numbers

Question 17

In a ring R, an element a is called a unit if:

Accepted Answer

There exists an element b in R such that ab = 1 and ba = 1.

Answer

a is the additive identity of R.

Answer

a is not invertible under addition.

Answer

a has an additive inverse.

Question 18

In a ring R, if a divides b, which of the following is true?

Accepted Answer

There exists an element c in R such that b = ac.

Answer

b is a multiple of a.

Answer

a and b are units in R.

Answer

a is a multiple of b.

Question 19

Which of the following is a necessary condition for a ring to be considered a field?

Accepted Answer

Every non-zero element is a unit.

Answer

The ring possesses a multiplicative identity.

Answer

The ring is commutative.

Answer

The ring is an integral domain.

Question 20

What is the formal definition of a ring?

Accepted Answer

A non-empty set with two binary operations, addition and multiplication, satisfying specific properties including the associative, commutative, and distributive properties.

Answer

A field with relaxed closure properties

Answer

A vector space with additional ring operations

Answer

A group with an additional multiplication operation

Question 21

Which property is not necessarily true for all rings?

Accepted Answer

Commutativity of multiplication

Answer

Existence of an additive identity

Answer

Distributivity of multiplication over addition

Answer

Associativity of addition

Question 22

In the ring of integers, what is the additive identity element?

Accepted Answer

0 (zero)

Answer

1 (one)

Answer

Any non-zero integer

Answer

-1 (negative one)

Question 23

In the ring of integers modulo 7, what is the multiplicative inverse of 3?

Accepted Answer

5

Answer

6

Answer

Inverse does not exist

Answer

2

Question 24

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

Accepted Answer

The set of all polynomials with coefficients in the field of real numbers

Answer

The set of all vectors in a finite-dimensional space

Answer

The set of all matrices with real entries

Answer

The set of all rational functions

Question 25

What is the degree of the polynomial f(x) = x^3 + 2x^2 - 5?

Accepted Answer

3 (cubic)

Answer

0 (constant)

Answer

2 (quadratic)

Answer

1 (linear)

Question 26

Which operation is not defined for all polynomial rings?

Accepted Answer

Division

Answer

Multiplication

Answer

Addition

Answer

Subtraction

Question 27

When the polynomial f(x) = x^3 + 2x^2 - 5 is divided by x - 1, what is the remainder?

Accepted Answer

-2

Answer

0

Answer

1

Answer

x - 1

Question 28

Define a ring.

Accepted Answer

A non-empty set equipped with two binary operations, addition and multiplication, satisfying specific axioms.

Answer

A set of numbers that are closed under addition and subtraction

Question 29

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

Accepted Answer

The set of real numbers under addition and multiplication

Answer

The set of polynomials with real coefficients under addition and multiplication

Answer

The set of integers under addition and multiplication

Question 30

Give an example of a non-commutative ring.

Accepted Answer

The ring of matrices with real entries

Answer

The ring of polynomials over a field

Answer

The ring of complex numbers

Answer

The ring of integers

Question 31

Provide an example of a finite field.

Accepted Answer

The field with two elements, {0, 1}

Answer

The field of rational numbers

Answer

The field of complex numbers

Question 32

Which of the following is NOT a part of the definition of a ring?

Accepted Answer

Commutative property of multiplication

Answer

Existence of multiplicative inverse

Answer

Existence of additive identity

Answer

Associative property of addition

Question 33

Consider the set of polynomials with complex coefficients. Which of the following statements is true?

Accepted Answer

It forms a ring with unity under polynomial addition and multiplication.

Answer

It does not form a ring under polynomial addition and multiplication.

Answer

It forms a field under polynomial addition and multiplication.

Question 34

Which of the following is a necessary condition for a ring to be a field?

Accepted Answer

Every nonzero element has a multiplicative inverse.

Answer

The ring is commutative under addition.

Answer

The ring has unity.

Question 35

Which of the following is NOT a requirement for a set with two binary operations to be considered a ring?

Accepted Answer

Multiplicative inverse for every non-zero element

Answer

Commutativity of addition

Answer

Associativity of addition

Answer

Existence of additive and multiplicative identities

Question 36

What is the additive identity element in the ring (ℤ₂,⊕₂,⊗₂), where ⊕₂ represents addition modulo 2 and ⊗₂ represents multiplication modulo 2?

Accepted Answer

0

Answer

1

Answer

2

Question 37

Which of the following rings is also a field?

Accepted Answer

ℚ

Answer

ℤ

Answer

M₂(ℝ)

Answer

ℤ₄

Question 38

What is the characteristic of the ring (?₆,⊕₆,⊗₆)?

Accepted Answer

6

Answer

5

Answer

3

Answer

2

Question 39

Which of the following is NOT a required property of rings?

Accepted Answer

Associativity of multiplication

Answer

Commutativity of addition

Answer

Existence of an additive identity

Answer

Distributivity of multiplication over addition

Question 40

In the ring M2(R), where R is a ring, what is the size of the matrix units?

Accepted Answer

2 x 2

Answer

3 x 3

Answer

1 x 1

Question 41

In the ring Z/12Z, what is the multiplicative inverse of 7?

Accepted Answer

5

Answer

12

Answer

7

Answer

11

Question 42

In the ring Z[x], what is the ideal generated by the polynomial x^2 - 1?

Accepted Answer

<x^2 - 1>

Answer

<x>

Answer

<x - 1>

Answer

<x + 1>

Question 43

Which of the following operations is not closed in the ring of integers?

Accepted Answer

Division

Answer

Subtraction

Answer

Addition

Answer

Multiplication

Question 44

What is the additive inverse of an element 'a' in a ring?

Accepted Answer

-a

Answer

a^2

Answer

0

Answer

1/a

Question 45

Which of the following is NOT a property of rings?

Accepted Answer

Distributive property over division

Answer

Associative property of addition

Answer

Existence of an additive identity

Answer

Commutative property of multiplication

Question 46

In the ring of polynomials over the field F, what is the multiplicative identity element?

Accepted Answer

1

Answer

0

Answer

x

Answer

x^2

Question 47

What is the characteristic of the ring of integers modulo n?

Accepted Answer

n

Answer

2

Answer

0

Answer

1

Question 48

Which of the following rings is a field?

Accepted Answer

Integers modulo 2

Answer

Integers

Answer

Ring of matrices

Answer

Polynomials over a field

Question 49

What is the order of the ring Z/6Z?

Accepted Answer

6

Answer

3

Answer

2

Answer

12

Question 50

Which of the following statements about rings is true?

Accepted Answer

Every ring has a unique zero element

Answer

Every ring is commutative

Answer

Every ring has a multiplicative inverse for every element

Answer

Every ring is a field