Question

The number of ideals in the ring \(\mathbb{Z}_{37}\) is

• A. \(4\)
• B. \(3\)
• C. \(2\)
• D. \(1\)

Hint:

A field has exactly two ideals: the zero ideal and the whole field.

Answer 1

The Correct Option is C

Concept:
If \(p\) is prime, then \[ \mathbb{Z}_p \] is a field. A field has only two ideals: \[ \{0\} \] and the whole field itself.

Step 1: Check whether \(37\) is prime. The number \(37\) is prime. Therefore, \[ \mathbb{Z}_{37} \] is a field.

Step 2: Ideals of a field. In a field \(F\), if an ideal contains any non-zero element \(a\), then since \(a\) has inverse \(a^{-1}\), we get \[ a^{-1}a=1 \] So the ideal contains \(1\), and hence it contains every element of the field. Therefore, the only ideals are \[ \{0\} \] and \[ F \]

Step 3: Final answer. Thus, the number of ideals in \(\mathbb{Z}_{37}\) is \[ \boxed{2} \]