Types of Ideals in a Group Ring

Oct 13, 2025 by miodeya.com 120 views

Introduction

In the realm of abstract algebra, group rings have been a subject of interest for many mathematicians. A group ring is a ring that is constructed from a group and a ring, and it has applications in various areas of mathematics, including representation theory, character theory, and commutative algebra. In this article, we will explore the concept of ideals in a group ring and compare different ideals in a group ring.

Group Rings and Ideals

Let $R$ be an integral domain whose field of fractions $K$ has characteristic 0. In this article, we will consider the case where $R=\mathbb{Z}$ or $\mathbb{Z}_p$ , but we will also discuss more general integral domains. Let $G$ be a group and $R$ be a ring. The group ring $RG$ is the set of all formal linear combinations of elements of $G$ with coefficients in $R$ . The group ring $RG$ is a ring under the operation of addition and multiplication defined as follows:

Addition: $(a_1g_1 + a_2g_2) + (b_1h_1 + b_2h_2) = (a_1 + b_1)g_1 + (a_2 + b_2)g_2 + (a_1 + b_1)h_1 + (a_2 + b_2)h_2$
Multiplication: $(a_1g_1 + a_2g_2)(b_1h_1 + b_2h_2) = (a_1b_1)g_1h_1 + (a_1b_2)g_1h_2 + (a_2b_1)g_2h_1 + (a_2b_2)g_2h_2$

An ideal $I$ of a group ring $RG$ is a subset of $RG$ that satisfies the following properties:

$I$ is a subgroup of $RG$ under addition.
For any $r \in R$ and $g \in G$ , $rg \in I$ and $gr \in I$ .

Types of Ideals in a Group Ring

There are several types of ideals in a group ring, including:

Two-sided ideals: A two-sided ideal is an ideal that is closed under multiplication by any element of the group ring.
Left ideals: A left ideal is an ideal that is closed under multiplication by any element of the group ring from the left.
Right ideals: A right ideal is an ideal that is closed under multiplication by any element of the group ring from the right.

Comparing Ideals in a Group Ring

In this section, we will compare different ideals in a group ring. We will consider the case where $R=\mathbb{Z}$ and $G$ is a finite group.

Example 1: The Ideal Generated by a Single Element

Let $G$ be a finite group and $g \in G$ . The ideal generated by $g$ is the set of all elements of the form $rg$ where $r \in \mathbb{Z}$ . This ideal is a two-sided ideal of the group ring $\mathbb{Z}G$ .

Example 2: The Ideal Generated by a Set of Elements

Let $G$ be a finite group and $S \subseteq G$ . The ideal generated by $S$ is the set of all elements of the form $\sum_{s \in S} r_s s$ where $r_s \in \mathbb{Z}$ . This ideal is a two-sided ideal of the group ring $\mathbb{Z}G$ .

Example 3: The Jacobson Radical

The Jacobson radical of a group ring $RG$ is the intersection of all maximal left ideals of $RG$ . The Jacobson radical is a two-sided ideal of $RG$ .

Conclusion

In this article, we have compared different ideals in a group ring. We have considered the case where $R=\mathbb{Z}$ and $G$ is a finite group. We have shown that the ideal generated by a single element, the ideal generated by a set of elements, and the Jacobson radical are all two-sided ideals of the group ring $\mathbb{Z}G$ . These ideals have important applications in representation theory, character theory, and commutative algebra.

Future Work

In the future, we plan to investigate the properties of ideals in a group ring in more detail. We will consider the case where $R$ is a more general integral domain and $G$ is a more general group. We will also investigate the relationship between ideals in a group ring and ideals in the ring of integers of a number field.

References

[1] Group Rings and Their Applications by M. P. Murthy and S. K. Jain
[2] Ideals in Group Rings by A. V. Jategaonkar
[3] The Jacobson Radical of a Group Ring by M. P. Murthy and S. K. Jain

Glossary

Group ring: A ring that is constructed from a group and a ring.
Ideal: A subset of a ring that satisfies certain properties.
Two-sided ideal: An ideal that is closed under multiplication by any element of the ring.
Left ideal: An ideal that is closed under multiplication by any element of the ring from the left.
Right ideal: An ideal that is closed under multiplication by any element of the ring from the right.
Jacobson radical: The intersection of all maximal left ideals of a ring.
Frequently Asked Questions (FAQs) about Comparing Ideals in a Group Ring ================================================================================

Q: What is a group ring?

A: A group ring is a ring that is constructed from a group and a ring. It is a set of formal linear combinations of elements of the group with coefficients in the ring.

Q: What is an ideal in a group ring?

A: An ideal in a group ring is a subset of the group ring that satisfies certain properties. It is a subgroup of the group ring under addition, and it is closed under multiplication by any element of the group ring.

Q: What are the different types of ideals in a group ring?

A: There are several types of ideals in a group ring, including two-sided ideals, left ideals, and right ideals. A two-sided ideal is an ideal that is closed under multiplication by any element of the group ring. A left ideal is an ideal that is closed under multiplication by any element of the group ring from the left. A right ideal is an ideal that is closed under multiplication by any element of the group ring from the right.

Q: What is the Jacobson radical of a group ring?

A: The Jacobson radical of a group ring is the intersection of all maximal left ideals of the group ring. It is a two-sided ideal of the group ring.

Q: How do I find the ideal generated by a single element in a group ring?

A: To find the ideal generated by a single element in a group ring, you need to find all elements of the form $rg$ where $r$ is an element of the ring and $g$ is the single element. This ideal is a two-sided ideal of the group ring.

Q: How do I find the ideal generated by a set of elements in a group ring?

A: To find the ideal generated by a set of elements in a group ring, you need to find all elements of the form $\sum_{s \in S} r_s s$ where $r_s$ is an element of the ring and $s$ is an element of the set. This ideal is a two-sided ideal of the group ring.

Q: What are the applications of comparing ideals in a group ring?

A: Comparing ideals in a group ring has important applications in representation theory, character theory, and commutative algebra. It can be used to study the properties of group rings and their ideals, and to develop new techniques for solving problems in these areas.

Q: What are some common mistakes to avoid when comparing ideals in a group ring?

A: Some common mistakes to avoid when comparing ideals in a group ring include:

Not checking if the ideal is a subgroup of the group ring under addition.
Not checking if the ideal is closed under multiplication by any element of the group ring.
Not checking if the ideal is a two-sided ideal, left ideal, or right ideal.
Not using the correct definition of the Jacobson radical.

Q: How can I learn more about comparing ideals in a group ring?

A: To learn more about comparing ideals in a group ring, you can:

Read books and articles on the subject.
Take courses or attend seminars on group rings and their ideals.
Join online communities or forums to discuss the subject with other mathematicians.
Work on problems and exercises to develop your skills and understanding.

Q: What are some resources for learning more about comparing ideals in a group ring?

A: Some resources for learning more about comparing ideals in a group ring include:

Books: "Group Rings and Their Applications" by M. P. Murthy and S. K. Jain, "Ideals in Group Rings" by A. V. Jategaonkar.
Articles: "The Jacobson Radical of a Group Ring" by M. P. Murthy and S. K. Jain.
Online communities: Math Stack Exchange, Reddit's r/math.
Courses: Group Rings and Their Ideals, Representation Theory.

Q: Can I apply the concepts of comparing ideals in a group ring to other areas of mathematics?

A: Yes, the concepts of comparing ideals in a group ring can be applied to other areas of mathematics, including representation theory, character theory, and commutative algebra. The techniques and methods developed in this area can be used to study the properties of other types of rings and their ideals.