Week 4 - Day 2: Rings and their Properties

What is a Ring?

A ring is a set equipped with two operations, addition and multiplication, satisfying several key properties that mirror arithmetic in the integers \( \mathbb{Z} \).

Addition Properties

  • Closure: For all \( a, b \in \mathbb{Z} \), the sum \( a + b \in \mathbb{Z} \).
  • Associativity: For all \( a, b, c \in \mathbb{Z} \), we have \( (a + b) + c = a + (b + c) \).
  • Identity Element: There exists an element \( 0_R \in \mathbb{Z} \) such that for all \( a \in \mathbb{Z} \), we have \( a + 0_R = a \).
  • Inverse Elements: For each \( a \in \mathbb{Z} \), there exists \( -a \in \mathbb{Z} \) such that \( a + (-a) = 0_R \).
  • Commutativity: For all \( a, b \in \mathbb{Z} \), we have \( a + b = b + a \).

Multiplication Properties

  • Closure: For all \( a, b \in \mathbb{Z} \), the product \( ab \in \mathbb{Z} \).
  • Associativity: For all \( a, b, c \in \mathbb{Z} \), we have \( (ab)c = a(bc) \).

Distributive Property

For all \( a, b, c \in \mathbb{Z} \), we have:

\[ a(b + c) = ab + ac \quad \text{and} \quad (b + c)a = ba + ca. \]

These are the essential properties that define a ring structure.

Associativity of Addition

We can add three numbers together, and associativity ensures that the order in which we add them doesn’t matter:

\[ a + (b + c) = (a + b) + c = a + (b + c). \]

Proposition: Uniqueness of Additive Identity

A ring can only have one additive identity element.

Proof

Suppose \( e \) and \( E \) are both additive identities. Then:

\[ e = e + E = E. \]

Therefore, the two identities must be equal. \(\blacksquare\)

Notation for Identity Elements

If \( R \) is a ring, we typically call the additive identity the zero element and denote it by \( 0_R \) instead of \( e \).

Proposition: Uniqueness of Additive Inverses

Each element in a ring can only have one additive inverse.

Proof

Suppose \( b \) and \( B \) are both additive inverses for \( a \). Then:

\[ b = b + 0_R = b + (a + B) = (b + a) + B = 0_R + B = B. \]

Therefore, the two inverses must be equal. \(\blacksquare\)

Notation for Additive Inverses

The additive inverse of an element \( a \) in a ring is denoted by \( -a \) rather than \( b \) or any other letter.

Examples of Rings

  • Even integers: \( 2\mathbb{Z} \) represents all even integers. This works for any \( n\mathbb{Z} \).
  • The integers: \( \mathbb{Z} \).
  • 2x2 real matrices: \( M_2(\mathbb{R}) \), the set of 2x2 matrices with real entries.
  • Rational numbers: \( \mathbb{Q} \).
  • Polynomials with real coefficients: \( \mathbb{R}[x] \), the set of all polynomials in one variable with coefficients from \( \mathbb{R} \).

Question

If \( R = M_2(\mathbb{R}) \), what is the additive identity \( 0_R \)?

Answer: \( 0_R = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \).

Multiplicative Identity

The integers \( \mathbb{Z} \) have another important property: the existence of a multiplicative identity.

There exists an element \( 1_R \in \mathbb{Z} \) such that for all \( a \in \mathbb{Z} \), we have:

\[ a \cdot 1_R = a = 1_R \cdot a. \]

Fact

A ring \( R \) can have at most one multiplicative identity, often denoted as \( 1_R \) rather than \( i \).

Question

If \( R = M_2(\mathbb{R}) \), what is the multiplicative identity \( 1_R \)?

Answer: \( 1_R = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \).

Definition: Ring with Identity

A ring that has a multiplicative identity is called a ring with identity.

Question

Which of the example rings does not have an identity?

Answer: \( 2\mathbb{Z} \) does not have a multiplicative identity.

Commutative Multiplication

The integers \( \mathbb{Z} \) have another property: commutative multiplication. For all \( a, b \in \mathbb{Z} \), we have:

\[ a \cdot b = b \cdot a. \]

Definition: Commutative Ring

A commutative ring is a ring where multiplication is commutative.

Question

Which of the example rings is not commutative?

Answer: \( M_2(\mathbb{R}) \) is not commutative. For example:

\[ \begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}, \]

but:

\[ \begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} 2 & 0 \\ 0 & 0 \end{bmatrix}. \]

Zero Factor Property (ZFP)

The integers \( \mathbb{Z} \) have another important property called the Zero Factor Property (ZFP). For all \( a, b \in \mathbb{Z} \), we have:

\[ \text{if } ab = 0 \text{, then } a = 0 \text{ or } b = 0. \]

Question

Which example does not satisfy the Zero Factor Property?

Answer: \( M_2(\mathbb{R}) \) does not satisfy the ZFP. For example:

\[ \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}, \]

but neither of the matrices are the zero matrix individually.

Definition: Integral Domain

A nonzero commutative ring with identity is called an integral domain if it satisfies the Zero Factor Property.

Question

Which of the examples are not integral domains?

  • \( M_2(\mathbb{R}) \), since it doesn't satisfy the ZFP and isn't commutative.
  • \( 2\mathbb{Z} \), since it doesn't have an identity.

Conclusion

The integers \( \mathbb{Z} \) are an integral domain, since they satisfy all the properties discussed.