Rings and Fields: Definitions and Elementary Properties
Discrete structures encompass mathematical objects that are countable or have distinct, separated values. Among these structures, rings and fields hold significant importance. Let's delve into their definitions, elementary properties, and examples.
Definition of Rings
A ring is an algebraic structure consisting of a set equipped with two binary operations, usually addition and multiplication, that satisfy specific properties.
- Additive Closure: For any two elements (a, b) in the ring, their sum a + b is also in the ring.
- Additive Associativity: Addition is associative, meaning (a + b) + c = a + (b + c) for all a, b, c in the ring.
- Additive Identity: There exists an element 0 in the ring such that a + 0 = a for all a in the ring.
- Additive Inverse: For every element a in the ring, there exists an element -a in the ring such that a + (-a) = 0.
- Multiplicative Closure: For any two elements a, b in the ring, their product ab is also in the ring.
- Multiplicative Associativity: Multiplication is associative, meaning (ab)c = a(bc) for all a, b, c in the ring.
- Distributive Property: Multiplication distributes over addition, i.e., a(b + c) = ab + ac and (a + b)c = ac + bc for all a, b, c in the ring.
Elementary Properties of Rings
- A ring may or may not have a multiplicative identity. If it does, it's called a unital ring or a ring with unity.
- Multiplication in a ring is not required to have inverses, unlike addition.
- If multiplication in a ring is commutative, i.e., ab = ba for all a, b in the ring, it's termed as a commutative ring.
Examples of Rings
Let's consider some examples:
- The set of integers Z with usual addition and multiplication forms a ring.
- The set of polynomials with real coefficients, denoted by R[x], with addition and multiplication of polynomials as operations, is a ring.
- Matrices over a field F form a ring under matrix addition and multiplication.
Definition of Fields
A field is a more structured algebraic system than a ring. It's a set equipped with two binary operations, addition and multiplication, such that certain properties hold.
- A field is a commutative ring with multiplicative inverses for all non-zero elements.
- Every non-zero element in a field has a multiplicative inverse. That is, for every a in the field, there exists b such that ab = 1.
- The field contains at least two elements, namely the additive identity (0) and the multiplicative identity (1), and these two elements are distinct.
Elementary Properties of Fields
- Multiplication and addition in a field are both commutative and associative.
- In a field, the distributive property holds.
- Fields do not have zero divisors, i.e., if ab = 0 in a field, then either a = 0 or b = 0.
Examples of Fields
Common examples of fields include:
- The set of rational numbers Q with usual addition and multiplication forms a field.
- The set of real numbers R and the set of complex numbers C are both fields.
- The set of integers modulo a prime number p, denoted by Zp, forms a field.