Ring Theory Quizzes: Test Your Understanding of Rings and Ideals

Question 1

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

Accepted Answer

The ring of matrices over a field.

Answer

The ring of integers.

Answer

The ring of polynomials over a field.

Answer

The ring of continuous functions on a compact interval.

Question 2

Which property is NOT essential for a mathematical structure to be considered a ring?

Accepted Answer

Multiplication is commutative

Answer

Addition is associative

Answer

There exists an additive identity element

Answer

Multiplication distributes over addition

Question 3

Which property uniquely defines an integral domain?

Accepted Answer

The product of any two non-zero elements is non-zero.

Answer

Every non-zero element has a multiplicative inverse.

Answer

Every ideal in the integral domain is a maximal ideal.

Question 4

Consider a quotient ring R/I. Which statement about its properties is correct?

Accepted Answer

Each ideal in R/I corresponds to a unique ideal in R.

Answer

Every quotient ring R/I is a field.

Answer

Every quotient ring R/I is isomorphic to the original ring R.

Answer

The order of R/I is always the same as the order of the ideal I.

Question 5

Which of the following statements holds true for all rings?

Accepted Answer

Every ring has a distinct zero element.

Answer

Every ring has a unity element.

Answer

Every ring is commutative.

Answer

Every ring consists of an infinite number of elements.

Question 6

In a ring R, which condition is equivalent to a + b = 0?

Accepted Answer

a = -b

Answer

a = 0 and b = 0

Answer

ab = 0

Answer

a - b = 0

Question 7

Let R be a ring, and I and J be ideals of R. Which statement about these ideals is always true?

Accepted Answer

The intersection I ∩ J is always an ideal of R.

Answer

The sum I + J is always an ideal of R.

Answer

The product I × J is always an ideal of R.

Answer

If I is a maximal ideal, then J must also be a maximal ideal.

Question 8

Which of the following conditions is **NOT** necessary for a ring to be a field?

Accepted Answer

Every nonzero element has an additive inverse

Answer

Associative addition and multiplication

Answer

Additive and multiplicative identity elements

Answer

Distributive property of multiplication over addition

Question 9

Which of the following is a field extension of Z/3Z?

Accepted Answer

Z/9Z

Answer

Z/5Z

Answer

Z/7Z

Answer

Z/11Z

Question 10

What is the order of the group of units of the ring Z/15Z?

Accepted Answer

6

Answer

4

Answer

8

Answer

12

Question 11

What is the characteristic of the ring Z/5Z[x]?

Accepted Answer

5

Answer

0

Answer

2

Answer

10

Question 12

Which of the following is a Galois extension?

Accepted Answer

Q(i)/Q

Answer

Q(√2)/Q

Answer

Z/4Z(√2)

Answer

Z/6Z(i)

Question 13

Let R be a ring and I be an ideal of R. What is the quotient ring R/I?

Accepted Answer

The set of all cosets a + I, where a ∈ R

Answer

The set of all ideals J of R such that I ⊆ J

Answer

The set of all elements in R that are not in I

Answer

The set of all subrings of R that contain I

Question 14

Which of the following is NOT a property of a field extension?

Accepted Answer

The field extension is an integral domain extension.

Answer

The field extension is a ring extension.

Answer

The field extension preserves the algebraic operations of addition and multiplication.

Answer

The field extension preserves the multiplicative identity element.

Question 15

Let F be a field and K be an algebraic extension of F. Which of the following statements is true?

Accepted Answer

F is a subfield of K.

Answer

K is a vector space over F.

Answer

K is a maximal ideal of F.

Answer

K is a quotient ring of F.

Question 16

What is the degree of the polynomial ring R[x] over the ring R?

Accepted Answer

∞

Answer

1

Answer

2

Answer

The degree is not defined for polynomial rings.

Question 17

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

Accepted Answer

The ring of matrices over a field

Answer

The ring of integers

Answer

The ring of real numbers

Answer

The ring of polynomials over a field

Question 18

Let R be a ring and I an ideal of R. Which of the following is always valid?

Accepted Answer

I is a group under addition

Answer

I is a subring of R

Answer

The quotient ring R/I is a field

Answer

I contains the multiplicative identity of R

Question 19

Let F be a field and K an extension field of F. Which condition is required for K to be a finite extension of F?

Accepted Answer

K's degree over F is finite

Answer

K is a vector space over F

Answer

K possesses a primitive element over F

Answer

K is algebraic over F

Question 20

Which of the following exemplifies a division ring?

Accepted Answer

The set of complex numbers

Answer

Z

Answer

Q

Answer

M2(R)

Question 21

Let R be a ring with unity. Which of the following is equivalent to asserting that R is commutative?

Accepted Answer

The multiplication of any two elements of R commutes

Answer

Every element of R is self-inverse

Answer

The identity element of R is unique

Answer

The collection of all invertible elements of R forms a group under multiplication

Question 22

Which of the following structures is NOT a ring?

Accepted Answer

The set of polynomials with real coefficients and addition and composition of functions

Answer

The set of integers with addition and multiplication

Answer

The set of real numbers with addition and multiplication

Answer

The set of rational numbers with addition and multiplication

Question 23

Which of the following is an ideal of the ring ℤ?

Accepted Answer

The set of all even integers

Answer

The set of all odd integers

Answer

The set of all factors of 6

Answer

The set of all prime numbers

Question 24

Which of the following rings is a field?

Accepted Answer

ℤ/nℤ where n is a prime number

Answer

ℤ

Answer

The set of 2x2 matrices with real entries

Answer

The set of polynomials with rational coefficients

Question 25

Which of the following is a defining condition for a ring (R, +, *) to be a field?

Accepted Answer

Every nonzero element of R has a multiplicative inverse.

Answer

R has an additive identity element.

Answer

R is commutative with respect to multiplication.

Answer

Every element of R can be represented as a sum of two other elements in R.

Question 26

An ideal I of a ring (R, +, *) is considered a proper ideal if:

Accepted Answer

I is a non-empty subset of R that is distinct from R.

Answer

I is a subring of R.

Answer

I is a product of all ideals in R.

Answer

I is contained within every other ideal of R.

Question 27

The Chinese Remainder Theorem asserts that for integers a, b, c and moduli m1, m2 and m3 such that m1, m2 and m3 are pairwise coprime, the system of congruences:

Accepted Answer

Has a unique solution modulo m1m2m3.

Answer

Has no solutions.

Answer

Has infinitely many solutions.

Answer

Has a unique solution modulo m1.

Question 28

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

Accepted Answer

Commutativity of multiplication.

Answer

Associativity of addition.

Answer

Distributivity of multiplication over addition.

Answer

Existence of a multiplicative identity element.

Question 29

Which of the following is not a method for constructing rings?

Accepted Answer

Vector space

Answer

Direct product of rings

Answer

Quotient ring

Answer

Polynomial ring

Question 30

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

Accepted Answer

Commutativity under multiplication

Answer

Existence of a multiplicative identity

Answer

Associativity under multiplication

Answer

Existence of an additive inverse for each element

Question 31

Within the ring Z, which of the following is a prime ideal?

Accepted Answer

(p), where p is a prime number

Answer

(2)

Answer

(10)

Answer

(6)

Question 32

In a ring R, an element 'a' is termed a(n) ________ if the ideal generated by 'a' coincides with the entire ring R.

Accepted Answer

unit

Answer

zero divisor

Answer

idempotent

Question 33

Identify an example of a non-commutative ring:

Accepted Answer

The ring of 2x2 matrices over a field R, denoted as M2(R)

Answer

The ring of integers, denoted as Z

Answer

The ring of complex numbers, denoted as C

Answer

The ring of polynomials with coefficients in R, denoted as R[x]

Question 34

Which of the following definitions accurately describes a homomorphism between two rings R and S?

Accepted Answer

A function f: R → S that preserves the ring operations (addition and multiplication)

Answer

A function f: R → S that is one-to-one

Answer

A function f: R → S that is onto

Question 35

The Chinese Remainder Theorem offers a method to solve which type of system of equations?

Accepted Answer

Systems of linear congruences

Answer

Systems of exponential equations

Answer

Systems of quadratic equations

Answer

Systems of polynomial equations

Question 36

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

Accepted Answer

Commutativity of multiplication

Answer

Existence of multiplicative identity

Answer

Existence of additive identity

Answer

Associativity of addition

Question 37

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

Accepted Answer

The ring of integers modulo 5, denoted as ℤ/5ℤ

Answer

The ring of polynomials with real coefficients

Answer

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

Answer

The ring of even integers

Question 38

Which of the following theorems is used to characterize prime and maximal ideals?

Accepted Answer

Krull's Theorem

Answer

The Chinese Remainder Theorem

Answer

The Fundamental Theorem of Algebra

Answer

The Sylow Theorems

Question 39

A ring R is called a division ring if:

Accepted Answer

Every nonzero element of R has a multiplicative inverse

Answer

Every element of R has a multiplicative inverse

Answer

Every ideal of R is maximal

Question 40

Which of the following is an example of a field, which is a commutative ring with unity in which every nonzero element has a multiplicative inverse?

Accepted Answer

The field of rational numbers, ℚ

Answer

The ring of continuous functions on the real line

Answer

The ring of polynomials with rational coefficients

Answer

The ring of integers, ℤ

Question 41

A ring homomorphism is a map between two rings that:

Accepted Answer

Preserves the operations of addition and multiplication

Answer

Is always injective

Answer

Preserves the order of the elements

Question 42

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

Accepted Answer

Every ideal is finitely generated

Answer

Every subring is Noetherian

Answer

Every quotient ring is Noetherian

Answer

Every ring homomorphism is injective

Question 43

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

Accepted Answer

Commutativity of multiplication

Answer

Existence of both additive and multiplicative identities

Answer

Associativity of multiplication

Answer

Associativity of addition

Question 44

What is the quotient ring of R by the ideal I?

Accepted Answer

The set of equivalence classes of elements of R modulo I

Answer

The set of ideals of R that contain I

Answer

The set of elements of R that are not in I

Question 45

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

Accepted Answer

Every non-zero element has a multiplicative inverse

Answer

The ring is commutative

Answer

The ring has a unit element

Question 46

What is the characteristic of a ring?

Accepted Answer

The smallest positive integer n such that n·1_R = 0_R, or 0 if no such integer exists

Answer

The number of elements in the ring

Answer

The largest element in the ring

Question 47

Which of the following is a theorem in ring theory?

Accepted Answer

The Chinese Remainder Theorem

Answer

The Fundamental Theorem of Algebra

Answer

The Pythagorean Theorem

Answer

The Prime Number Theorem

Question 48

What is the relationship between a ring and its ideals?

Accepted Answer

The set of ideals of a ring forms a lattice under the operations of intersection and addition

Answer

The set of ideals of a ring forms a group

Answer

The set of ideals of a ring forms a vector space

Answer

The set of ideals of a ring forms a field