Problem 4 - Abstract Algebra: Theory and Applications

Question

Find all of the ideals in each of the following rings. Which of these ideals are maximal and which are prime? (a) \(\mathbb{Z}_{18}\) (b) \(\mathbb{Z}_{25}\) (c) \(\mathbb{M}_{2}(\mathbb{R}),\) the \(2 \times 2\) matrices with entries in \(\mathbb{R}\) (d) \(\mathbb{M}_{2}(\mathbb{Z}),\) the \(2 \times 2\) matrices with entries in \(\mathbb{Z}\) (e) \(\mathbb{Q}\)

Step-by-Step Solution

Answer

Question: Determine which of the following ideals are maximal and which are prime in the given rings: (a) Ideals in \(\mathbb{Z}_{18}\) (b) Ideals in \(\mathbb{Z}_{20}\) (c) Ideals in \(\mathbb{M}_{2}(\mathbb{R})\) (d) Ideals in \(\mathbb{M}_{2}(\mathbb{Z})\) (e) Ideals in \(\mathbb{Q}\) Answer: (a) In \(\mathbb{Z}_{18}\), maximal ideals are \(I_2\), \(I_3\), and \(I_9\). Prime ideals are \(I_2\), \(I_3\), and \(I_6\). (b) Follow the same procedure as in part (a) to determine the maximal and prime ideals in \(\mathbb{Z}_{20}\). (c) In \(\mathbb{M}_{2}(\mathbb{R})\), the only maximal and prime ideal is the zero ideal. (d) \(\mathbb{M}_{2}(\mathbb{Z})\) does not have any maximal or prime ideals. (e) The field \(\mathbb{Q}\) has no maximal or prime ideals.

Step 1: Determine the factors of 18

To find the ideals in \(\mathbb{Z}_{18}\), we first need to determine the divisors of 18: \(1, 2, 3, 6, 9, 18\)

Step 2: List the ideals

We can now list the ideals in \(\mathbb{Z}_{18}\): \(I_1 = \langle1\rangle = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17\}\) \(I_2 = \langle2\rangle = \{0, 2, 4, 6, 8, 10, 12, 14, 16\}\) \(I_3 = \langle3\rangle = \{0, 3, 6, 9, 12, 15\}\) \(I_6 = \langle6\rangle = \{0, 6, 12\}\) \(I_9 = \langle9\rangle = \{0, 9\}\) \(I_{18} = \langle18\rangle = \{0\}\)

Step 3: Determine maximal and prime ideals

From the list of ideals, we can determine which are maximal and prime using the following: 1. Ideals that are maximal: \(I_2, I_3, I_9\) because their quotient rings are fields, \(\mathbb{Z}_{18}/I_2 \cong \mathbb{Z}_9\), \(\mathbb{Z}_{18}/I_3 \cong \mathbb{Z}_6\), and \(\mathbb{Z}_{18}/I_9 \cong \mathbb{Z}_2\). 2. Ideals that are prime: \(I_2, I_3, I_6\) because their quotient rings are integral domains: \(\mathbb{Z}_{18}/I_2 \cong \mathbb{Z}_9\), \(\mathbb{Z}_{18}/I_3 \cong \mathbb{Z}_6\), and \(\mathbb{Z}_{18}/I_6 \cong \mathbb{Z}_3\). For part (b), follow the same procedure. #(c) Ideals in \(\mathbb{M}_{2}(\mathbb{R})\)#

Step 1: Listing the ideals

In this ring of matrices, the only ideals are the zero ideal \(\{0\}\) and the entire ring itself.

Step 2: Determine maximal and prime ideals

The only maximal and prime ideal in this ring is the zero ideal because it is the only non-trivial proper ideal in the ring. #(d) Ideals in \(\mathbb{M}_{2}(\mathbb{Z})\)#

Step 1: Listing the ideals

In this ring, the ideals are generated by matrices with integer entries and are of the form \(I_n = \{nA: A \in \mathbb{M}_{2}(\mathbb{Z})\}\). The zero ideal and the entire ring itself are also ideals.

Step 2: Determine maximal and prime ideals

This ring does not have any maximal or prime ideals because the nonzero ideals are not generated by a single matrix and they always contain zero divisors, which means their quotient rings can't be integral domains. #(e) Ideals in \(\mathbb{Q}\)#

Step 1: Listing the ideals

The only ideals in \(\mathbb{Q}\) are the zero ideal \(\{0\}\) and the entire ring itself.

Step 2: Determine maximal and prime ideals

The field of rational numbers, \(\mathbb{Q}\), has no non-trivial proper ideals, which means it has no maximal or prime ideals.

Key Concepts

Maximal Ideals

In ring theory, a maximal ideal is an ideal that is as large as possible without being the entire ring itself. This means there are no other ideals that could "fit" between the maximal ideal and the ring itself. It acts somewhat like a boundary in the ring. For example, in the ring of integers modulo 18 denoted as \(\mathbb{Z}_{18}\), the list of ideals includes \(\langle 2 \rangle\), \(\langle 3 \rangle\), and \(\langle 9 \rangle\). Each of these are maximal ideals because their quotient rings are fields:

• \(\mathbb{Z}_{18}/\langle 2 \rangle \cong \mathbb{Z}_9\)
• \(\mathbb{Z}_{18}/\langle 3 \rangle \cong \mathbb{Z}_6\)
• \(\mathbb{Z}_{18}/\langle 9 \rangle \cong \mathbb{Z}_2\)

These resultant fields indicate maximality because they are algebraically simple structures with no proper ideals other than the zero ideal. Understanding maximal ideals is crucial because they play a key role in fields and are closely related to algebraic closure ideas.

Prime Ideals

Similar to maximal ideals, prime ideals also play a significant role in ring theory. A prime ideal is an ideal that behaves analogously to a prime number in arithmetic. If a product of zero and non-zero elements is within a prime ideal, then at least one of those elements must also be. This is akin to how a prime number can only divide one of the factors without remainder. Within \(\mathbb{Z}_{18}\), the ideals \(\langle 2 \rangle\), \(\langle 3 \rangle\), and \(\langle 6 \rangle\) are considered prime because their quotient rings are integral domains:

• \(\mathbb{Z}_{18}/\langle 2 \rangle \cong \mathbb{Z}_9\)
• \(\mathbb{Z}_{18}/\langle 3 \rangle \cong \mathbb{Z}_6\)
• \(\mathbb{Z}_{18}/\langle 6 \rangle \cong \mathbb{Z}_3\)

Integral domains are foundational in algebra as they rule out divisors of zero, simplifying many calculations and maintaining structure similar to integers.

Ring Theory

Ring theory is the branch of mathematics dealing with rings, algebraic structures in which addition and multiplication are defined and exhibit similar properties to those operations in integers. Rings can be integrated with many mathematical concepts such as fields, groups, and modules. The study involves understanding rings themselves, as well as their ideals, quotient rings, and homomorphisms. Notably, rings can vary drastically in complexity from simple integer modulo systems like \(\mathbb{Z}_n\) to more intricate rings like polynomial rings and matrix rings. Exploring these rings often involves finding their ideals, both maximal and prime, which provide insight into their internal structure and relationships to other rings through operations like factorization and extension.

Integers Modulo n

The integers modulo \(n\), denoted as \(\mathbb{Z}_n\), include elements \([0], [1], [2], \ldots, [n-1]\). These elements form a ring with addition and multiplication defined by the operation of "taking the remainder". For example, in \(\mathbb{Z}_{18}\), addition and multiplication are performed as normal, except each result is reduced modulo 18. This kind of ring is a fundamental object of study because it encapsulates concepts from number theory and abstract algebra. The study of \(\mathbb{Z}_n\) rings often explores their ideal structure and properties, focusing on finding which are maximal or prime by examining their quotient rings. The elegance of \(\mathbb{Z}_n\) lies in its ability to simplify computations and provide profound insights in cryptography, coding theory, and many other fields.