How to Determine if M_{2×2}R over Ring Z is a Module

Introduction

Working with mathematical structures, such as rings and modules, is fundamental in many areas of mathematics, including algebra and abstract algebra. One crucial question is to determine if a set, in this case, the set of 2×2 matrices with entries in a ring R, can be regarded as a module over the ring of integers, mathbb{Z}.

Understanding the Problem

Let's consider the set M_{2×2}R, which represents the set of all 2×2 matrices with entries from a ring R. Our goal is to determine if this set is a module over mathbb{Z}. In other words, we want to check if M_{2×2}R satisfies the necessary conditions to be a mathbb{Z}-module.

Key Concepts

Ring

A ring R is a set equipped with two binary operations (addition and multiplication) that satisfy certain axioms. These axioms include the existence of an additive identity (0), a multiplicative identity (1 if the ring is unital), and distributivity of multiplication over addition.

Module

A mathbb{Z}-module, also known as a mathbb{Z}-vector space, is an abelian group (M, ) together with an action of the integers on M that satisfies the following properties for all (a, b in mathbb{Z}) and (m, n in M): a(n m) an am (a b)n an bn 1n n (-1)n -n

2×2 Matrices

A matrice of the form [begin{pmatrix} a b c d end{pmatrix}] where (a, b, c, d in R), is an element of M_{2×2}R. The set of all such matrices forms a ring under the operations of addition and multiplication defined component-wise.

Approach to the Problem

Step 1: Understanding Ring (M_{2×2}R)

First, notice that M_{2×2}R itself is a ring. It is a ring under the operations of matrix addition and matrix multiplication, both defined component-wise. Matrix addition is associative and commutative, and the zero matrix acts as the additive identity. Matrix multiplication is associative, and the identity matrix acts as a multiplicative identity.

Step 2: Forming a mathbb{Z}-Module

To show that M_{2×2}R is a mathbb{Z}-module, we need to check that the set of 2×2 matrices with entries in (R) forms an abelian group under addition and that there is an action of (mathbb{Z}) on this group that satisfies the module axioms.

The set of 2×2 matrices with entries in (R) does indeed form an abelian group under addition, as addition of matrices is commutative and associative, with the zero matrix serving as the identity element. Matrix addition also satisfies the inverse property, as adding a matrix to its negative results in the zero matrix.

Step 3: Action of (mathbb{Z}) on (M_{2×2}R)

The action of (mathbb{Z}) on M_{2×2}R can be defined by scalar multiplication. For a scalar (n in mathbb{Z}) and a matrix (A in M_{2×2}R), the product (nA) is defined as:

[n begin{pmatrix} a b c d end{pmatrix} begin{pmatrix} na nb nc nd end{pmatrix}]

We need to verify that this action satisfies the module axioms: n(m A) nA nB: This follows from the distributivity of scalar multiplication over matrix addition. (n m)A nA mA: This also follows from the distributivity of scalar multiplication over matrix addition. (mn)A m(nA): This follows from the associative property of scalar multiplication. 1A A: This is clear as the scalar 1 acts as the multiplicative identity. (-1)A -A: This is clear as (-1) acts as the additive inverse.

Conclusion

Based on the above discussion, it is clear that the set of 2×2 matrices with entries in (R), denoted M_{2×2}R, can indeed be regarded as a mathbb{Z}-module. Therefore, the answer to the question is that M_{2×2}R is a mathbb{Z}-module.

Exercise

For a more solid understanding, you are encouraged to prove that any ring (S) is a mathbb{Z}-module. This exercise will strengthen your understanding of the underlying algebraic structures and module theory.

Further Reading

If you are interested in delving deeper into this topic, consider exploring the following resources:

Module (mathematics) on Wikipedia for a comprehensive overview. Books like "Abstract Algebra" by David S. Dummit and Richard M. Foote for detailed discussions on rings and modules. Online lectures or courses on abstract algebra and module theory.