Properties of Rings: Master the Fundamentals of Ring Theory

Question 1

Which condition must be satisfied for a non-zero ring R to be a division ring?

Accepted Answer

Every non-zero element of R has a multiplicative inverse

Answer

R is commutative under multiplication

Answer

R is commutative under addition

Question 2

A ring in which every element commutes with every other element is called a:

Accepted Answer

Commutative ring

Answer

Division ring

Answer

Non-commutative ring

Answer

Integral domain

Question 3

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

Accepted Answer

The ring of matrices over the complex numbers

Answer

The ring of polynomials

Answer

The ring of real numbers

Answer

The ring of integers

Question 4

An element a in a ring (R,+,*) is called idempotent if:

Accepted Answer

a^2 = a

Answer

a^2 = 0

Answer

a*a = a

Answer

a^2 = 1

Question 5

Let R be a ring and a, b be elements of R. Which of the following implies that a * b = 0?

Accepted Answer

a = 0 or b = 0.

Answer

a - b = 0.

Answer

a + b = 0.

Answer

a * b is not equal to 0.

Question 6

Which of the following is NOT a defining property of a ring?

Accepted Answer

Existence of multiplicative inverse for every non-zero element

Answer

Associativity of multiplication

Answer

Distributivity of multiplication over addition

Question 7

Let R be a ring with unity. If a, b ∈ R and ab = 0, then which statement is true?

Accepted Answer

Either a = 0 or b = 0

Answer

a - b = 0

Answer

ab ≠ 0

Answer

a + b = 0

Question 8

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

Accepted Answer

The ring of even integers

Answer

The set of rational numbers

Answer

The set of complex numbers

Answer

The set of prime numbers

Question 9

In a ring R, the set of elements that commute with every other element in R is called the:

Accepted Answer

Center of R

Answer

Normalizer

Answer

Centralizer

Answer

Commutator subgroup

Question 10

Which of the following statements about ideals in a ring R is true?

Accepted Answer

The intersection of any two ideals of R is an ideal of R

Answer

The union of any two ideals of R is an ideal of R

Answer

Every ideal of R is a subring of R

Question 11

Which of the following is an example of a finite integral domain?

Accepted Answer

The ring of integers modulo 5

Answer

The ring of continuous functions on the interval [0, 1]

Answer

The ring of rational numbers

Answer

The ring of polynomials over the real numbers

Question 12

Which of the following is a defining property of a ring?

Accepted Answer

Associativity of both addition and multiplication, and distributivity of multiplication over addition

Answer

Only distributivity is a property of rings

Answer

Only associativity is a property of rings

Question 13

Which of the following is NOT a property of multiplication in a ring?

Accepted Answer

Commutativity

Answer

Associativity

Answer

Distributivity over addition

Question 14

In a ring with unity, what is the multiplicative inverse of 0?

Accepted Answer

Undefined

Answer

The element itself

Answer

0

Answer

1

Question 15

In a ring with zero divisors, what is a defining characteristic?

Accepted Answer

The product of two non-zero elements can be zero

Answer

The ring has a multiplicative identity

Answer

The ring is commutative

Answer

The product of two non-zero elements is always non-zero

Question 16

Which of the following statements about ideals in rings is true?

Accepted Answer

An ideal is a non-empty subset that is closed under addition and multiplication by elements of the ring

Answer

An ideal is a non-empty subset that is closed under multiplication by elements of the ring

Question 17

Which of the following is an example of a field?

Accepted Answer

The set of rational numbers (Q) under addition and multiplication

Answer

The set of even numbers under addition and multiplication

Answer

The set of integers (Z) under addition and multiplication

Question 18

In a ring, the cancellation law for addition states that:

Accepted Answer

For any elements a, b, and c, if a + b = a + c, then b = c

Answer

For any elements a, b, and c, if a * b = a * c, then b = c

Answer

For any elements a and b, if a + b = 0, then a = b = 0

Question 19

Identify the property of rings that is NOT included in the following list:

Accepted Answer

Associativity of subtraction

Answer

Associativity of multiplication

Answer

Distributivity of multiplication over addition

Answer

Additive identity

Question 20

Which set forms a subring of the ring of integers?

Accepted Answer

The set of even integers

Answer

The set of odd integers

Answer

The set of rational numbers

Answer

The set of positive integers

Question 21

Which statement accurately describes a fundamental property of rings?

Accepted Answer

Every ring possesses at least one zero element.

Answer

Every ring inevitably has an identity element.

Answer

All rings are inherently commutative.

Question 22

Which ring best exemplifies the characteristics of a field?

Accepted Answer

The field of complex numbers

Answer

The ring of integers

Answer

The ring of matrices over a field

Answer

The ring of polynomials over a field

Question 23

Which condition is necessary for a ring to qualify as an integral domain?

Accepted Answer

The absence of zero divisors within the ring

Answer

The commutative property of the ring

Answer

The associative property of the ring

Answer

The presence of a multiplicative identity in the ring

Question 24

Which of the following is NOT a property of rings?

Accepted Answer

Commutativity of addition

Answer

Associativity of multiplication

Answer

Distributivity of multiplication over addition

Answer

Existence of additive identity

Question 25

Let R be a ring. Which of the following statements is equivalent to the condition that R is commutative?

Accepted Answer

For all elements a and b in R, the product of a and b is equal to the product of b and a, i.e., ab = ba

Answer

For every element a in R, there exists an element b in R such that the product of a and b is equal to the zero element, i.e., ab = 0

Answer

For every element a in R, the square of a is equal to a, i.e., a^2 = a

Answer

For all elements a and b in R, the sum of a and b is equal to the sum of b and a, i.e., a + b = b + a

Question 26

Consider the ring of integers, denoted as Z. Which of the following sets forms a subring of Z?

Accepted Answer

The set of even integers

Answer

The set of rational numbers

Answer

The set of odd integers

Answer

The set of real numbers

Question 27

Let R be a ring with a unity element. For an element a in R, which of the following conditions is equivalent to a being a unit in the ring?

Accepted Answer

There exists an element b in R such that the product of a and b, and the product of b and a, both result in the unity element, i.e., ab = ba = 1

Answer

a is not equal to the zero element of R

Answer

There exists an element b in R such that the product of a and b, and the product of b and a, both result in the zero element, i.e., ab = ba = 0

Answer

a has an additive inverse in R

Question 28

Which of the following rings is a field?

Accepted Answer

The ring of complex numbers, denoted as C

Answer

The ring of polynomials with real number coefficients

Answer

The ring of integers, denoted as Z

Answer

The ring of matrices with real number entries

Question 29

Consider a ring R that does not possess any zero divisors. Which of the following statements is a valid consequence of this property?

Accepted Answer

Every non-zero element of R is a unit

Answer

R is a field

Answer

R is commutative

Answer

R has a unity element

Question 30

In the context of the ring of integers, Z, which of the following sets qualifies as an ideal?

Accepted Answer

The set of even integers

Answer

The set of rational numbers

Answer

The set of real numbers

Answer

The set of odd integers

Question 31

Let R be a ring with a unity element and I be an ideal in R. Which statement accurately describes an ideal?

Accepted Answer

I is a subring of R

Answer

I is commutative

Answer

I is a field

Answer

I has a unity element

Question 32

Which of the following functions qualifies as a ring homomorphism from Z, the ring of integers, to itself?

Accepted Answer

f(a) =2a

Answer

f(a) = |a|

Answer

f(a) = 1/a

Answer

f(a) = a^2

Question 33

In the context of ring theory, which function between the ring of integers, Z, and the ring of even integers, 2Z, is considered an isomorphism?

Accepted Answer

f(a) = 2a

Answer

f(a) =a^2

Answer

f(a) = |a|

Question 34

Which of the following is NOT a property of rings?

Accepted Answer

Distributivity of addition over multiplication

Answer

Associativity of multiplication

Answer

Existence of additive and multiplicative identities

Answer

Associativity of addition

Question 35

Let R be a ring and let a, b ∈ R. Which of the following is true?

Accepted Answer

(a + b)² = a² + 2ab + b²

Answer

(a + b)² = 2a² + 2b²

Answer

(a + b)² = a² - 2ab + b²

Answer

(a + b)² = a² + b²

Question 36

Let R be a ring and let a ∈ R. Which of the following is true?

Accepted Answer

-a = -1 * a

Answer

-a = a²

Answer

-a = 1 * a

Answer

-a = 0

Question 37

Let R be a ring with characteristic 5. Which of the following is true?

Accepted Answer

5a = 0 for all a ∈ R

Answer

5 is the only prime number in R

Answer

The additive identity in R is 5

Answer

R is a field

Question 38

Let R be a ring and let a, b ∈ R. Which of the following is true?

Accepted Answer

If a² = 0, then a = 0 or R has zero divisors

Answer

If a² = 0, then ab = 0 for all b ∈ R

Answer

If a² = 0, then a is a unit in R

Question 39

Which of the following is NOT a property of rings?

Accepted Answer

Commutativity of multiplication

Answer

Distributivity of multiplication over addition

Answer

Associativity of addition

Answer

Existence of a multiplicative identity

Question 40

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

Accepted Answer

1 is the only multiplicative identity in R.

Answer

0 can also be a multiplicative identity in R.

Answer

There can be multiple multiplicative identities in R.

Answer

Every non-zero element of R is a multiplicative identity.

Question 41

Consider the following binary operation on the set Z of integers: a * b = a + b - 1. Does this operation define a ring?

Accepted Answer

Yes

Answer

No, because the operation does not have a multiplicative identity.

Answer

No, because the operation is not associative.

Question 42

Which of the following is an example of a ring with more than one multiplicative identity?

Accepted Answer

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

Answer

The set of integers under addition

Answer

The set of rational numbers under multiplication

Question 43

Let R be a ring with unity. Which of the following is true?

Accepted Answer

The additive inverse of every element in R is unique.

Answer

Every element in R has both an additive and multiplicative inverse.

Answer

The multiplicative inverse of every non-zero element in R is unique.

Question 44

Consider the following statement: "Every ring is a group under addition."

Accepted Answer

True

Answer

False, because rings do not have an identity element for addition.

Answer

False, because rings allow multiplication, which is not a group operation.

Answer

False, because rings do not have an inverse element for every element under addition.

Question 45

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

Accepted Answer

Every non-zero element has a multiplicative inverse.

Answer

The ring must have a unity element.

Answer

The ring must be commutative under multiplication.

Question 46

Consider the following statement: "Every field is a ring."

Accepted Answer

True

Answer

False, because fields allow division, which is not a ring operation.

Answer

False, because fields do not have an additive identity.

Answer

False, because fields do not have a multiplicative identity.

Question 47

Which of the following is an example of an infinite ring?

Accepted Answer

The set of polynomials with real coefficients under addition and multiplication

Answer

The set of rational numbers under addition and multiplication

Answer

The set of integers under addition and multiplication

Question 48

Consider the following statement: "Every ring with unity is a field."

Accepted Answer

False

Answer

True, because every ring with unity has an additive inverse for every element.

Answer

True, because every ring with unity has a unique additive and multiplicative identity.

Answer

True, because every ring with unity has a multiplicative inverse for every non-zero element.