Problem 4 - A Book of Abstract Algebra

Question

Show that the set of all \(2 \times 2\) matrices of the form $$ \left(\begin{array}{ll} 0 & 0 \\ 0 & x \end{array}\right) $$ is a subring of \(\mathscr{M}_{2}(\mathbb{R})\), then prove this subring is isomorphic to \(\mathbb{R}\) For any integer \(k\), let \(k \mathbb{Z}\) designate the subring of \(\mathbb{Z}\) which consists of all the multiples of \(k\).

Step-by-Step Solution

Answer

The set \( S \) is a subring of \( \mathscr{M}_{2}(\mathbb{R}) \) and is isomorphic to \( \mathbb{R} \).

Step 1: Introduction to the Problem

We need to show that the set of all matrices of a certain form is a subring of \( \mathscr{M}_{2}(\mathbb{R}) \) and that it is isomorphic to \( \mathbb{R} \). The matrices are of the form \( \left(\begin{array}{ll} 0 & 0 \ 0 & x \end{array}\right) \).

Step 1: Define Set of Matrices

Let \( S = \left\{ \left( \begin{array}{cc} 0 & 0 \ 0 & x \end{array} \right) \mid x \in \mathbb{R} \right\} \). We must show \( S \) is a subring of \( \mathscr{M}_{2}(\mathbb{R}) \).

Step 2: Check Closure under Addition

For any matrices \( A = \left( \begin{array}{cc} 0 & 0 \ 0 & x \end{array} \right) \) and \( B = \left( \begin{array}{cc} 0 & 0 \ 0 & y \end{array} \right) \), the sum \( A + B = \left( \begin{array}{cc} 0 & 0 \ 0 & (x+y) \end{array} \right) \) is also of the same form. Therefore, \( S \) is closed under addition.

Step 3: Check Closure under Multiplication

The product \( A \times B = \left( \begin{array}{cc} 0 & 0 \ 0 & x \end{array} \right) \times \left( \begin{array}{cc} 0 & 0 \ 0 & y \end{array} \right) = \left( \begin{array}{cc} 0 & 0 \ 0 & xy \end{array} \right) \), which is of the same form. Thus, \( S \) is closed under multiplication.

Step 4: Check Existence of Additive Identity

The zero matrix \( \left( \begin{array}{cc} 0 & 0 \ 0 & 0 \end{array} \right) \) serves as the additive identity and is in \( S \).

Step 5: Check Existence of Additive Inverses

For a matrix \( A = \left( \begin{array}{cc} 0 & 0 \ 0 & x \end{array} \right) \), its additive inverse is \( -A = \left( \begin{array}{cc} 0 & 0 \ 0 & -x \end{array} \right) \), which is in \( S \).

Step 6: Subring Verification Conclusion

Since \( S \) is closed under addition and multiplication, and contains the additive identity and inverses, \( S \) is a subring of \( \mathscr{M}_{2}(\mathbb{R}) \).

Step 7: Verify Isomorphism to \( \mathbb{R} \)

Define a map \( \phi: S \rightarrow \mathbb{R} \) by \( \phi\left( \begin{array}{cc} 0 & 0 \ 0 & x \end{array} \right) = x \). This map is bijective and preserves addition and multiplication, making \( S \) isomorphic to \( \mathbb{R} \).

Step 8: Verification of the Map as a Homomorphism

We check that the map \( \phi \) preserves operations: \( \phi(A + B) = \phi\left( \begin{array}{cc} 0 & 0 \ 0 & x+y \end{array} \right) = x + y = \phi(A) + \phi(B) \) and \( \phi(AB) = \phi\left( \begin{array}{cc} 0 & 0 \ 0 & xy \end{array} \right) = xy = \phi(A) \cdot \phi(B) \). Thus, \( \phi \) is a homomorphism.

Step 9: Conclusion of Isomorphism

Since \( \phi \) is bijective and a homomorphism, \( S \) is isomorphic to \( \mathbb{R} \).

Step 10: Analyze \( k \mathbb{Z} \) in Context

For the context part of the exercise, note that \( k\mathbb{Z} \) can be seen as the set of integers that are multiples of \( k \), forming another type of subring within the integers.

Key Concepts

Subring

In abstract algebra, a **subring** is a subset of a ring that is itself a ring with respect to the same operations. To prove a set is a subring, it must satisfy several criteria:

• Closure under addition and multiplication: For any elements in the subset, the sum and product must also be in the subset.
• Containment of the additive identity: The zero element (additive identity) of the larger ring must be in the subset.
• Existence of additive inverses: For each element in the subset, there must be an additive inverse also within the subset.

In the example from the exercise, we dealt with matrices of the form \( \left( \begin{array}{cc} 0 & 0 \ 0 & x \end{array} \right) \). This subset was shown to be closed under the operations of addition and multiplication. The zero matrix served as the additive identity, and every matrix had an inverse that remained within the subset. Thus, this subset was verified to be a subring of \( \mathscr{M}_{2}(\mathbb{R}) \), the set of all \( 2 \times 2 \) real matrices.

Isomorphism

An **isomorphism** between two algebraic structures shows that they are structurally the same, even if they appear different. It involves a bijective function (one-to-one and onto) between sets that preserves the operations of the structure.

In our exercise, after establishing the subset of matrices as a subring, we demonstrated it was isomorphic to the set of real numbers \( \mathbb{R} \). The mapping \( \phi: S \rightarrow \mathbb{R} \) defined by \( \phi \left( \begin{array}{cc} 0 & 0 \ 0 & x \end{array} \right) = x \) was bijective and preserved addition as well as multiplication. For example, \( \phi(A + B) \) equaled \( \phi(A) + \phi(B) \), fulfilling both the bijectivity and the operational preservation, necessary for an isomorphism. Being isomorphic to \( \mathbb{R} \) confirms that the subring behaves, in terms of its structure, like the real numbers.

Matrix Algebra

**Matrix algebra** refers to performing algebraic operations with matrices. It closely resembles working with numbers, but extra rules apply due to matrices being multidimensional arrays.

In the textbook exercise, the set of matrices given had specific and simple forms: zero elements in three positions and a real number in the lower right corner. This specificity simplified challenges observed otherwise in matrix algebra, allowing straightforward verification of subring properties.

Matrices are added by adding corresponding elements, and similarly, matrices are multiplied considering the dot-product method, ensuring dimension compatibility. In the 2x2 matrices provided in the exercise, the operations were simplified as the multiplicative and additive behaviors mimicked that of real numbers, given their structure.

Ring Homomorphism

A **ring homomorphism** is a function between two rings that respects the ring operations (addition and multiplication). It demonstrates the structured relationship between two rings.

For a function \( \phi \) to be a ring homomorphism, it must satisfy:

• \( \phi(a + b) = \phi(a) + \phi(b) \)
• \( \phi(a \cdot b) = \phi(a) \cdot \phi(b) \)

In our problem, the function \( \phi \) from the subring to \( \mathbb{R} \) fulfilled these criteria, preserving the operations within \( S \) and \( \mathbb{R} \). This property of \( \phi \) ensured not only that \( S \) acted like \( \mathbb{R} \) structurally but also that it maintained the same relational properties and algebraic behavior through this homomorphism, further supported by its bijective nature.