Problem 11 - A First Course in Module Theory

Question

Let \(R\) be the set of all infinite matrices \(A=\left(a_{i j}\right)\) over a field \(F\), with rows and columns indexed by the positive integers, subject to the condition that each row and column of \(A\) has only a finite number of nonzero entries. Verify that \(R\) is a ring under the expected rules of addition and multiplication. Find a set of left ideals \(I_{1}, \ldots\) so that \(R\) is an internal direct sum \(\oplus I_{i}\) as a left \(R\)-module. Let \(m R\) be the set of all matrices \(A\) in \(R\) that have only a finite number of nonzero entries. Show that \(m R\) is a two-sided ideal of \(R\). (This provides a contrast to Exercise \(7.8\) above.)

Step-by-Step Solution

Answer

R is a ring under the given rules. \( R = \oplus I_i \) where \( I_i \) are ideals based on rows. \( mR \) is a two-sided ideal.

Step 1: Define the Set and Conditions

Let’s start by defining the set \( R \) of infinite matrices \( A = (a_{ij}) \) over a field \( F \). Each row and column of these matrices must have only a finite number of nonzero entries. This will be the basis of our discussion.

Step 2: Verify Ring Properties

To verify that \( R \) is a ring, we need to check two main properties: closure under addition and closure under multiplication.\(\)- **Addition**: If \( A = (a_{ij}) \) and \( B = (b_{ij}) \) are elements of \( R \), their sum \( C = A + B \) such that \( c_{ij} = a_{ij} + b_{ij} \) should also be in \( R \). Since both \( A \) and \( B \) have finite nonzero entries per row and column, their sum will also be finite for rows and columns.\(\)- **Multiplication**: For multiplication, we use the rule \( C = A \times B \) where \( c_{ij} = \sum_{k} a_{ik} b_{kj} \). The finite nonzero nature in each row and column ensures the product remains in \( R \). Thus, \( R \) is closed under addition and multiplication.

Step 3: Define Left Ideals and Internal Direct Sum

We need to find a set of left ideals \( I_1, I_2, \ldots \) that form an internal direct sum of \( R \). One way to construct these ideals is to use matrices that have nonzero entries only on the i-th row or j-th column.\(\)- Define \( I_i \) as the set of matrices whose nonzero entries are only in the i-th row. Similarly, define \( J_j \) as matrices with nonzero entries only in the j-th column. Together, these ideals cover all matrices in \( R \).\(\)- Consequently, \( R \) can be expressed as an internal direct sum: \( R = \oplus_{i} I_i \).

Step 4: Show \( mR \) is a Two-Sided Ideal

Define \( mR \) as the set of matrices in \( R \) with only a finite number of nonzero entries, extending the concept from individual rows and columns to the whole matrix.\(\)- **Two Sided Ideal**: For \( A \in R \) and \( B \in mR \), consider the products \( AB \) and \( BA \). Since \( B \) has finite non-zero entries overall, the results \( AB \) and \( BA \) will also have only finite nonzero entries due to the multiplication rules creating row and column sums with finite contributions.\(\)- Thus, \( mR \) is closed under multiplication from elements in \( R \), proving \( mR \) is a two-sided ideal.

Key Concepts

Infinite Matrices

An infinite matrix is a matrix with an infinite number of rows and columns. However, for the set of matrices in this exercise, each row and column has only a finite number of nonzero entries. This restriction is crucial to ensuring certain properties of the set.When working with infinite matrices, operations like addition and multiplication follow similar rules to finite matrices. The primary difference lies in managing the infinity aspect, but the finite nonzero entries condition ensures those operations remain well-defined. Remember, the finite nonzero condition is what keeps this set manageable and suitable for forming a ring.

Field Theory

Field theory underpins the structure of the set of infinite matrices. A field is a set of elements satisfying certain algebraic rules, such as addition, subtraction, multiplication, and division (except by zero). Fields provide the scalars over which vector spaces, rings, and other algebraic structures are defined.In this context, the field provides the elements from which the entries of the infinite matrices are drawn. It ensures that any finite combinations of these entries via field operations (like those in summing or multiplying matrices) yield results within the same field.

Module Theory

Module theory extends the concept of vector spaces to more general operations. In a module, the elements can be scaled by elements of a ring, not just a field. The set of infinite matrices we are examining can be seen as a module over itself.To express these matrices as an internal direct sum, we partition it into smaller, more manageable pieces: the left ideals. A left ideal is a subset of a ring where any element of the subset multiplied by any element of the ring (from the left) is still within the subset. The exercise constructs left ideals using rows and columns, showing how the infinite set can be broken down into finite parts. This breakdown into an internal direct sum makes the module structure clearer and provides a modular approach to studying the ring.

Two-Sided Ideal

A two-sided ideal in a ring is a subset that remains within the subset after multiplication by any element of the ring from both sides. In this exercise, the set of matrices with only a finite number of nonzero entries, denoted as 'mR', is shown to be a two-sided ideal.This means that if you take any matrix from 'mR' and multiply it by any matrix from the ring (either from the left or the right), the result will also be a matrix from 'mR'. This property is vital for the analysis and decomposition of ring structures, as it allows 'mR' to serve as a kind of building block for more complex constructions within the ring.