Introduction to Ring Theory

Mathematics is built upon abstract structures that generalize arithmetic operations. Among these, rings play a fundamental role, appearing in number theory, algebraic geometry, and cryptography. In this article, we will introduce ring theory, from axioms to homomorphisms, along with the Fundamental Theorem on Ring Homomorphisms.

1. Definition of a Ring

A ring is a set \( R \) equipped with two binary operations, addition (\( + \)) and multiplication (\( \cdot \)), satisfying the following axioms:

Axioms of a Ring

• Additive Associativity: \( a + (b + c) = (a + b) + c \) for all \( a, b, c \in R \).
• Additive Commutativity: \( a + b = b + a \) for all \( a, b \in R \).
• Additive Identity (Zero Element): There exists an element \( 0 \in R \) such that \( a + 0 = a \).
• Additive Inverses: For every \( a \in R \), there exists \( -a \) such that \( a + (-a) = 0 \).
• Multiplicative Associativity: \( a \cdot (b \cdot c) = (a \cdot b) \cdot c \).
• Distributive Laws: \( a \cdot (b + c) = a \cdot b + a \cdot c \) and \( (a + b) \cdot c = a \cdot c + b \cdot c \).

A ring is commutative if \( a \cdot b = b \cdot a \) for all \( a, b \in R \).

A ring has an identity if there exists \( 1 \in R \) such that \( a \cdot 1 = 1 \cdot a = a \) for all \( a \in R \).

2. Examples of Rings

• Integers \( \mathbb{Z} \): The set of integers under usual addition and multiplication is a commutative ring with identity.
• Matrix Rings \( M_n(\mathbb{R}) \): The set of \( n \times n \) real matrices forms a ring but is not commutative for \( n \geq 2 \).
• Polynomial Rings \( \mathbb{R}[x] \): The set of all polynomials with real coefficients is a commutative ring.

3. Ring Homomorphisms

A ring homomorphism is a function \( \phi: R \to S \) between two rings that satisfies:

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

If \( \phi \) is bijective, it is called an isomorphism.

4. Kernels and Ideals

The kernel of a ring homomorphism is:

\[ \ker(\phi) = \{a \in R \mid \phi(a) = 0\}. \]

The kernel is always an ideal of \( R \), meaning it is closed under addition and absorbs multiplication from elements in \( R \).

5. Fundamental Theorem on Ring Homomorphisms

If \( \phi: R \to S \) is a ring homomorphism, then:

1. The kernel \( \ker(\phi) \) is an ideal of \( R \).
2. The image \( \phi(R) \) is a subring of \( S \).
3. The quotient ring \( R / \ker(\phi) \) is isomorphic to \( \phi(R) \): \[ R / \ker(\phi) \cong \phi(R). \]

Conclusion

Ring theory is fundamental in algebra, number theory, and cryptography. In this article, we covered:

• The axioms of rings
• Examples of rings
• Ring homomorphisms
• Fundamental Theorem on Ring Homomorphisms

In the next article, we will explore quotient rings, factorization, and Noetherian rings. Stay tuned!