Question

Consider the rings
\[ S_1 = \mathbb{Z}[x]/\langle 2, x^3 \rangle \quad \text{and} \quad S_2 = \mathbb{Z}_2[x]/\langle x^2 \rangle \]
where \(\langle 2, x^3 \rangle\) denotes the ideal generated by \(\{2, x^3\}\) in \(\mathbb{Z}[x]\) and \(\langle x^2 \rangle\) denotes the ideal generated by \(x^2\) in \(\mathbb{Z}_2[x]\).
Which of the following statements is/are correct?

• A. Every prime ideal of \(S_1\) is a maximal ideal
• B. \(S_2\) has exactly one maximal ideal
• C. Every element of \(S_1\) is either nilpotent or a unit
• D. There exists an element in \(S_2\) which is NEITHER nilpotent NOR a unit

Hint:

For a quotient ring \(k[x]/\langle p(x)^n \rangle\) where \(k\) is a field and \(p(x)\) is irreducible, the ring is a local ring. Its unique maximal ideal is \(\langle \overline{p(x)} \rangle\). The units are the elements not in this ideal, and the nilpotent elements are the elements in this ideal.

Answer 1

The Correct Option is A, C

Step 1: Understanding the Concept:
We need to analyze the properties of two quotient rings. The key is to understand the structure of these rings, including their elements, ideals, units, and nilpotent elements.
Step 3: Detailed Explanation:
Analysis of Ring \(S_1\):
First, we simplify the structure of \(S_1\). By the third isomorphism theorem for rings:
\[ S_1 = \mathbb{Z}[x]/\langle 2, x^3 \rangle \cong (\mathbb{Z}[x]/\langle 2 \rangle) / \langle \overline{x^3} \rangle \] Since \(\mathbb{Z}[x]/\langle 2 \rangle \cong \mathbb{Z}_2[x]\), we have \(S_1 \cong \mathbb{Z}_2[x]/\langle x^3 \rangle\).
The elements of \(S_1\) are the residue classes of polynomials in \(\mathbb{Z}_2[x]\) of degree less than 3. The general form is \(a_0 + a_1x + a_2x^2\) where \(a_i \in \mathbb{Z}_2\). The ring \(S_1\) is finite with \(2^3=8\) elements.
(A) Every prime ideal of \(S_1\) is a maximal ideal. In any finite commutative ring, every prime ideal is maximal. Since \(S_1\) is a finite ring, this statement is true. (A) is TRUE.
(C) Every element of \(S_1\) is either nilpotent or a unit. In a local ring like \(\mathbb{Z}_2[x]/\langle x^3 \rangle\), the non-units form the unique maximal ideal \(\langle x \rangle\). An element is in \(\langle x \rangle\) if its constant term is 0. An element is nilpotent if some power is 0. In this ring, \(p(x)^k \equiv 0 \pmod{x^3}\) is equivalent to \(x|p(x)\).
Thus, the nilpotent elements are precisely the non-units. An element \(p(x)\) is a unit if and only if its constant term is 1. It is nilpotent if and only if its constant term is 0. Since every element has a constant term of either 0 or 1, every element is either nilpotent or a unit. (C) is TRUE.
Analysis of Ring \(S_2\): The ring \(S_2 = \mathbb{Z}_2[x]/\langle x^2 \rangle\) consists of elements of the form \(a_0 + a_1x\) where \(a_i \in \mathbb{Z}_2\). It has \(2^2=4\) elements: \(\{0, 1, x, 1+x\}\).
(B) \(S_2\) has exactly one maximal ideal. A commutative ring is a local ring if it has a unique maximal ideal. The maximal ideals of \(S_2\) correspond to the maximal ideals of \(\mathbb{Z}_2[x]\) that contain \(\langle x^2 \rangle\).
The maximal ideals in \(\mathbb{Z}_2[x]\) are generated by irreducible polynomials. The only irreducible factor of \(x^2\) is \(x\). So the only maximal ideal of \(\mathbb{Z}_2[x]\) containing \(\langle x^2 \rangle\) is \(\langle x \rangle\). Thus, \(S_2\) has a unique maximal ideal, which is \(\langle \bar{x} \rangle = \{0, x\}\). (B) is TRUE.
(D) There exists an element in \(S_2\) which is NEITHER nilpotent NOR a unit. Let's check all non-zero elements:
- \(1\): A unit (\(1 . 1 = 1\)).
- \(x\): Nilpotent (\(x^2 = 0\)).
- \(1+x\): A unit (\((1+x)(1+x) = 1+2x+x^2 = 1+0+0 = 1\)).
All non-zero elements are either units or nilpotent. So the statement is false. (D) is FALSE.
Step 4: Final Answer:
The correct statements are (A), (B), and (C).