Ring Theory

A ring is a non-empty set R equipped with two binary operations, called addition (+) and multiplication (×), that satisfy the following axioms: 1. Associativity of addition: For all a, b, c ∈ R, (a + b) + c = a + (b + c). 2. Commutativity of addition: For all a, b ∈ R, a + b = b + a. 3. Existence of additive identity: There exists an element 0 ∈ R, called the additive identity, such that for all a ∈ R, a + 0 = a. 4. Existence of additive inverse: For each a ∈ R, there exists an element -a ∈ R, called the additive inverse of a, such that a + (-a) = 0. 5. Associativity of multiplication: For all a, b, c ∈ R, (a × b) × c = a × (b × c). 6. Distributivity of multiplication over addition: For all a, b, c ∈ R, a × (b + c) = (a × b) + (a × c).

The set of integers ℤ forms a ring under the usual addition and multiplication operations. It satisfies all the axioms of a ring, including associativity, commutativity, existence of additive identity (0), existence of additive inverse, associativity of multiplication, and distributivity of multiplication over addition.

A field is a ring in which every nonzero element has a multiplicative inverse. This means that for every nonzero element a ∈ F, there exists an element b ∈ F such that a × b = b × a = 1, where 1 is the multiplicative identity of the field.

The set of rational numbers ℚ forms a field under the usual addition and multiplication operations. It satisfies all the axioms of a field, including associativity, commutativity, existence of additive identity (0), existence of additive inverse, associativity of multiplication, distributivity of multiplication over addition, and the existence of multiplicative inverses for all nonzero elements.

An ideal in a ring R is a non-empty subset I of R that is closed under addition and multiplication. This means that for all a, b ∈ I and r ∈ R, a + b ∈ I and ra, ar ∈ I.

The set of even integers forms an ideal in the ring of integers ℤ. It is non-empty, closed under addition (the sum of two even integers is even), and closed under multiplication by an integer (the product of an even integer and an integer is even).

A prime ideal in a ring R is an ideal P that is not contained in any larger ideal other than itself and the whole ring R. In other words, if I is an ideal such that P ⊆ I ≠ I ∉ R, then P = I.

The set of prime numbers forms a prime ideal in the ring of integers ℤ. It is an ideal because it is non-empty, closed under addition, and closed under multiplication by an integer. It is prime because it is not contained in any larger ideal other than itself and the whole ring ℤ.

A maximal ideal in a ring R is an ideal M that is not contained in any larger ideal other than itself. In other words, if I is an ideal such that M ⊆ I ≠ I ∉ R, then M = I.

The set of even integers forms a maximal ideal in the ring of integers ℤ. It is an ideal because it is non-empty, closed under addition, and closed under multiplication by an integer. It is maximal because it is not contained in any larger ideal other than itself.

A ring homomorphism is a function f: R → S between two rings R and S that preserves the ring operations. This means that for all a, b ∈ R, f(a + b) = f(a) + f(b) and f(a × b) = f(a) × f(b).

The function f: ℤ → ℚ that sends each integer n to its rational representation n/1 is a ring homomorphism. It preserves the addition and multiplication operations: f(a + b) = (a + b)/1 = a/1 + b/1 = f(a) + f(b) and f(a × b) = (a × b)/1 = a/1 × b/1 = f(a) × f(b).

A ring isomorphism is a ring homomorphism that is one-to-one and onto. This means that for all a, b ∈ R, f(a) = f(b) if and only if a = b, and for every b ∈ S, there exists an a ∈ R such that f(a) = b.

The function f: ℤ → ℚ that sends each integer n to its rational representation n/1 is a ring isomorphism. It is one-to-one because if f(a) = f(b), then a/1 = b/1, which implies a = b. It is onto because for every rational number m/n, we can find an integer a = m such that f(a) = m/n.

The Chinese Remainder Theorem states that for any two relatively prime integers m and n, the system of congruences x ∈ m (mod m) and x ∈ n (mod n) has a unique solution modulo mn. This means that there exists an integer x such that x ∈ m (mod m) and x ∈ n (mod n), and this solution is unique modulo mn.