Question 1:
\[
\text{Which of the following is an integral domain but not a field?}
\]
\[
\text{(a) } \mathbb{Z}, \quad \text{(b) } \mathbb{Q}, \quad \text{(c) } \mathbb{R}, \quad \text{(d) } \mathbb{C}
\]
Answer: A
Question 2:
\[
\text{Let } R \text{ be a commutative ring with unity. If } R \text{ has no zero divisors, what is } R \text{ called?}
\]
\[
\text{(a) } A field, \quad \text{(b) } A division ring, \quad \text{(c) } An integral domain, \quad \text{(d) } A group
\]
Answer: C
Question 3:
\[
\text{Which of the following is NOT a property of an ideal in a commutative ring?}
\]
\[
\text{(a) } Closure under addition, \quad \text{(b) } Closure under multiplication by ring elements, \quad \text{(c) } Containing the multiplicative identity, \quad \text{(d) } Closure under additive inverses
\]
Answer: C
Question 4:
\[
\text{What is the characteristic of the ring } \mathbb{Z}/7\mathbb{Z} \text{?}
\]
\[
\text{(a) } 0, \quad \text{(b) } 1, \quad \text{(c) } 7, \quad \text{(d) } None of these
\]
Answer: C
Question 5:
\[
\text{Which of the following rings is a principal ideal domain (PID)?}
\]
\[
\text{(a) } \mathbb{Z}, \quad \text{(b) } \mathbb{Z}[x], \quad \text{(c) } \mathbb{Z}[\sqrt{2}], \quad \text{(d) } \mathbb{Q}[x]
\]
Answer: A
Question 6:
\[
\text{The nilradical of a commutative ring consists of:}
\]
\[
\text{(a) } Units, \quad \text{(b) } Zero divisors, \quad \text{(c) } Nilpotent elements, \quad \text{(d) } Prime elements
\]
Answer: C
Question 7:
\[
\text{Which of the following is NOT an example of a Noetherian ring?}
\]
\[
\text{(a) } \mathbb{Z}, \quad \text{(b) } \mathbb{Q}[x], \quad \text{(c) } \mathbb{R}[x_1, x_2, \dots], \quad \text{(d) } \mathbb{C}[x]
\]
Answer: C
Question 8:
\[
\text{A maximal ideal in a commutative ring is always:}
\]
\[
\text{(a) } A prime ideal, \quad \text{(b) } A principal ideal, \quad \text{(c) } A zero ideal, \quad \text{(d) } A unit
\]
Answer: A
Question 9:
\[
\text{If } R \text{ is a commutative ring with unity, the set of all units in } R \text{ is called:}
\]
\[
\text{(a) } An ideal, \quad \text{(b) } A group, \quad \text{(c) } A subring, \quad \text{(d) } A field
\]
Answer: B
Question 10:
\[
\text{Which of the following rings is a unique factorization domain (UFD)?}
\]
\[
\text{(a) } \mathbb{Z}, \quad \text{(b) } \mathbb{Z}[x], \quad \text{(c) } \mathbb{Q}[x], \quad \text{(d) } All of these
\]
Answer: D
Question 11:
\[
\text{A ring } R \text{ is called a local ring if it has exactly one:}
\]
\[
\text{(a) } Ideal, \quad \text{(b) } Maximal ideal, \quad \text{(c) } Prime ideal, \quad \text{(d) } Nilradical
\]
Answer: B
Question 12:
\[
\text{The ring } \mathbb{Z}/4\mathbb{Z} \text{ is:}
\]
\[
\text{(a) } A field, \quad \text{(b) } An integral domain, \quad \text{(c) } A ring with zero divisors, \quad \text{(d) } A Noetherian ring
\]
Answer: C
Question 13:
\[
\text{Which of the following statements is true?}
\]
\[
\text{(a) } Every PID is a UFD, \quad \text{(b) } Every UFD is a PID, \quad \text{(c) } Every integral domain is a PID, \quad \text{(d) } Every field is a UFD
\]
Answer: A
Question 14:
\[
\text{An ideal } I \text{ of a commutative ring } R \text{ is prime if:}
\]
\[
\text{(a) } ab \in I \Rightarrow a \in I \text{ or } b \in I, \quad \text{(b) } I = R, \quad \text{(c) } I \text{ is maximal}, \quad \text{(d) } I \text{ contains units}
\]
Answer: A
Question 15:
\[
\text{If } I \text{ is an ideal of } R, \text{ then } R/I \text{ is a field if and only if } I \text{ is:}
\]
\[
\text{(a) } A maximal ideal, \quad \text{(b) } A prime ideal, \quad \text{(c) } A principal ideal, \quad \text{(d) } A radical ideal
\]
Answer: A
Question 16:
\[
\text{The polynomial ring } \mathbb{Q}[x] \text{ is:}
\]
\[
\text{(a) } A PID, \quad \text{(b) } A UFD, \quad \text{(c) } A field, \quad \text{(d) } None of these
\]
Answer: B
Question 17:
\[
\text{Which of the following rings is NOT a domain?}
\]
\[
\text{(a) } \mathbb{Z}, \quad \text{(b) } \mathbb{Q}, \quad \text{(c) } \mathbb{Z}/6\mathbb{Z}, \quad \text{(d) } \mathbb{Q}[x]
\]
Answer: C
Question 18:
\[
\text{A Noetherian ring is one in which:}
\]
\[
\text{(a) } Every ideal is finitely generated, \quad \text{(b) } Every prime ideal is maximal, \quad \text{(c) } The ring is a UFD, \quad \text{(d) } None of these
\]
Answer: A
Question 19:
\[
\text{Which of the following is an example of an Artinian ring?}
\]
\[
\text{(a) } A field, \quad \text{(b) } \mathbb{Z}, \quad \text{(c) } \mathbb{Q}[x], \quad \text{(d) } None of these
\]
Answer: A
Question 20:
\[
\text{The radical of an ideal } I \text{ in } R \text{ consists of:}
\]
\[
\text{(a) } Nilpotent elements, \quad \text{(b) } Prime elements, \quad \text{(c) } Units, \quad \text{(d) } None of these
\]
Answer: A
