Multiple Choice Questions
Abstract Algebra: Groups and Rings
Topic: Groups
Grade: 11
Question 1: Which of the following is not a group under addition?
a) Integers
b) Rational numbers
c) Real numbers
d) Complex numbers
Answer: d) Complex numbers
Explanation: Complex numbers do not form a group under addition because not every complex number has an additive inverse. For example, the complex number 1 + i does not have an additive inverse within the set of complex numbers. However, all the other options (integers, rational numbers, and real numbers) do form groups under addition.
Example: Let\’s consider the set of integers. For any two integers a and b, their sum a + b is always an integer. Additionally, for every integer a, there exists an additive inverse -a such that a + (-a) = 0. Therefore, the set of integers forms a group under addition.
Question 2: Which of the following is a group under multiplication?
a) Positive real numbers
b) Non-zero real numbers
c) Non-zero rational numbers
d) Non-zero complex numbers
Answer: a) Positive real numbers
Explanation: The set of positive real numbers forms a group under multiplication because every positive real number has a multiplicative inverse. For example, the multiplicative inverse of 2 is 1/2, and their product is 1. The other options (non-zero real numbers, non-zero rational numbers, and non-zero complex numbers) do not form groups under multiplication because not every element has a multiplicative inverse.
Example: Let\’s consider the set of positive real numbers. For any two positive real numbers a and b, their product ab is always a positive real number. Additionally, for every positive real number a, there exists a multiplicative inverse 1/a such that a * (1/a) = 1. Therefore, the set of positive real numbers forms a group under multiplication.
Topic: Rings
Grade: 11
Question 3: Which of the following is not a commutative ring?
a) Integers
b) Rational numbers
c) Real numbers
d) Complex numbers
Answer: d) Complex numbers
Explanation: Complex numbers do not form a commutative ring because not every complex number has a multiplicative inverse. For example, the complex number 0 does not have a multiplicative inverse within the set of complex numbers. However, all the other options (integers, rational numbers, and real numbers) do form commutative rings.
Example: Let\’s consider the set of integers. For any two integers a and b, their sum a + b and product ab are always integers. Additionally, the set of integers is closed under addition and multiplication, associative, has an additive identity (0), and every element has an additive inverse. Therefore, the set of integers forms a commutative ring.
Question 4: Which of the following is an example of a non-commutative ring?
a) Integers
b) Rational numbers
c) Real numbers
d) Complex numbers
Answer: a) Integers
Explanation: The set of integers forms a non-commutative ring because under multiplication, the order of the elements matters. For example, 2 * 3 = 6, but 3 * 2 = 6 as well. The other options (rational numbers, real numbers, and complex numbers) form commutative rings because the order of multiplication does not affect the result.
Example: Let\’s consider the set of integers. For any two integers a and b, their sum a + b and product ab are always integers. Additionally, the set of integers is closed under addition and multiplication, associative, has an additive identity (0), and every element has an additive inverse. However, the set of integers is not commutative under multiplication, as shown by the example above. Therefore, the set of integers forms a non-commutative ring.