/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 8 In Exercises 1 through 10 determ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 8

In Exercises 1 through 10 determine whether the indicated set \(l\) is an ideal in the indicated ring \(R\). $$ I=\left\\{\left[\begin{array}{ll} 0 & n \\ 0 & m \end{array}\right] \mid n, m \in \mathbb{Z}\right\\} \text { in } R=\left\\{\left[\begin{array}{ll} a & b \\ 0 & c \end{array}\right] \mid a, b, c \in \mathbb{Z}\right\\} $$

Short Answer

Expert verified
The set \(I\) is an ideal in the ring \(R\).

Step by step solution

01

Define Ideals in Rings

An ideal \(I\) in a ring \(R\) is a subset such that for every element \(r\) in \(R\) and every element \(i\) in \(I\), both \(r \cdot i\) and \(i \cdot r\) are in \(I\). Also, \(I\) must be non-empty and closed under addition and multiplication by elements of \(R\).
02

Check Non-Empty Property

Verify that the given set \(I\) is non-empty. Since \(I\) consists of matrices of the form \(\begin{bmatrix} 0 & n \ 0 & m \end{bmatrix}\), it is clear that it includes the matrix \(\begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix}\). Hence, \(I\) is non-empty.
03

Closure Under Addition

Check if \(I\) is closed under addition. Consider two elements in \(I\): \(\begin{bmatrix} 0 & n_1 \ 0 & m_1 \end{bmatrix}\) and \(\begin{bmatrix} 0 & n_2 \ 0 & m_2 \end{bmatrix}\). Their sum is \(\begin{bmatrix} 0 & (n_1+n_2) \ 0 & (m_1+m_2) \end{bmatrix}\), which is also in \(I\). Hence, \(I\) is closed under addition.
04

Closure Under Multiplication by Elements of R

Check if \(I\) is closed under multiplication by elements of \(R\). Consider any element \(r = \begin{bmatrix} a & b \ 0 & c \end{bmatrix} \in R\) and \(i = \begin{bmatrix} 0 & n \ 0 & m \end{bmatrix} \in I\). Calculate \(r \cdot i = \begin{bmatrix} a \cdot 0 + b \cdot 0 & a \cdot n + b \cdot m \ 0 \cdot 0 + c \cdot 0 & 0 \cdot m + c \cdot m \end{bmatrix} = \begin{bmatrix} 0 & an + bm \ 0 & cm \end{bmatrix}\), which is an element of \(I\). Similarly, \(i \cdot r = \begin{bmatrix} 0 & a \cdot n + n \cdot c \ 0 & m \cdot c \end{bmatrix}\) is also in \(I\). This proves closure under multiplication by elements of \(R\).
05

Conclusion

Since \(I\) is non-empty, closed under addition, and closed under multiplication by any element of the ring \(R\), we conclude that \(I\) is an ideal in \(R\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ideals in Rings
In ring theory, an ideal is a special subset of a ring that has unique algebraic properties. Think of an ideal as a sort of sub-world within the ring where certain multiplication rules apply. Given a ring \( R \), an ideal \( I \) within this ring must satisfy specific criteria:
  • **Non-empty**: The ideal must contain at least one element, typically the zero element of the ring.
  • **Closed under addition**: For every pair of elements in the ideal, their sum must also lie in the ideal.
  • **Closed under multiplication by ring elements**: Multiplying any element of the ideal by any element of the ring must still produce an element in the ideal.
In the given exercise, the ideal \( I \) is a set of matrices defined within the broader ring \( R \). By ensuring that \( I \) is non-empty and satisfies closure under both addition and multiplication by elements of \( R \), we establish that \( I \) is indeed an ideal within \( R \). This framework ensures consistent properties that are critical for abstract algebra and applications in algebraic structures.
Matrix Algebra
Matrix algebra is a powerful and versatile tool in mathematics, particularly evident in the study of linear transformations and systems of linear equations. A matrix is a rectangular array of numbers arranged in rows and columns. In the context of ring theory, matrices can form rings themselves when we define specific operations on them.
  • Matrices can be added together or multiplied, following standard arithmetic rules for matrix operations.
  • In our exercise, both the ideal \( I \) and the ring \( R \) consist of matrices. Each matrix is defined with specific entries, which are integers, denoted as \( \mathbb{Z} \).
Matrix multiplication is especially significant as it highlights the idea of closure within the context of ideals. When we multiply a matrix from \( I \) with a matrix from \( R \), we confirm if the result remains within set \( I \). This property underpins the concept of operation consistency and is vital in many calculus and algebra applications, like transformations in geometry.
Closure Properties
Closure properties refer to the requirement that certain set operations must yield results within the same set. These are fundamental in the definition of algebraic structures like groups, rings, and fields. For the set \( I \) to be considered an ideal in ring \( R \), it must be closed under both addition and multiplication by any element of \( R \).
  • **Closure under addition**: If you add two matrices from set \( I \), the resulting matrix must also be in \( I \). This is checked by ensuring internal consistency in the operation, as addition should not produce elements outside the designated boundaries of \( I \).
  • **Closure under multiplication by ring elements**: This involves taking any element from the ring \( R \) and any element from \( I \) and verifying that their product still lies in \( I \). This ensures that no matter how the ideal interacts with the broader structure, it remains internally consistent.
By verifying these closure properties, we ensure that the operations within the ideal mirror those of the ring itself, maintaining the substructure's integrity. This consistency is crucial for the further study of algebraic properties and behaviors in mathematical systems.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In Exercises 1 through 15 determine the field of quotients of the indicated rings if it exists. If it does not exist, explain why. $$ \mathbb{Z} $$

In Exercises 1 through 15 determine the field of quotients of the indicated rings if it exists. If it does not exist, explain why. $$ Q(t)=\left\\{a+b i \mid a, b \in Q, i^{2}=-1\right\\} $$

Find all possible ring homomorphisms between the indicated rings. $$ \phi: 3 \mathbb{Z} \rightarrow \mathbb{Z} $$

In Exercises 1 through 10 determine whether the indicated set \(l\) is an ideal in the indicated ring \(R\). $$ I=\left\\{(2 x, 3 x) \mid x \in \mathbb{Z}_{6}\right\\} \text { in } R=\mathbb{Z}_{6} \times \mathbb{Z}_{6} $$

Let \(I\) and \(J\) be ideals in a ring \(R\). (a) Show that \(I+J=\\{a+b \mid a \in I, b \in J\\}\) is an ideal. (b) Show that \(I J=\left\\{a_{1} b_{1}+a_{2} b_{2}+\ldots+a_{n} b_{n} \mid a_{i} \in I, b_{i} \in J\right\\}\) is an ideal. (c) Show that \(I J \subseteq I \cap J\). (d) If \(R\) is commutative and \(I+J=R,\) show that \(I J=I \cap J .\)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.