A ring is a set of numbers (called elements) with two binary operations, typically denoted as addition (+) and multiplication (*), that satisfy the following properties:
Closure under addition and multiplication: For any elements a and b in the ring, the result of a + b and a * b are also elements of the ring.
Commutativity of addition: For any elements a and b in the ring, a + b = b + a.
Associativity of addition and multiplication: For any elements a, b, and c in the ring, the following are true: (a + b) + c = a + (b + c) and (a * b) * c = a * (b * c).
Identity element under addition: There exists an element 0 in the ring such that for every element a in the ring, a + 0 = a.
Inverse elements under addition: For every element a in the ring, there exists an element -a in the ring such that a + (-a) = 0.
Multiplicative identity: There exists an element 1 in the ring such that for every element a in the ring, a * 1 = a.
Distributivity of multiplication over addition: For any elements a, b, and c in the ring, a * (b + c) = a * b + a * c and (a + b) * c = a * c + b * c.
In addition to these properties, an integral domain is a ring where the only zero divisors are 0 and the additive identity, meaning that if a and b are non-zero elements of the ring and a * b = 0, then at least one of a or b must be 0.
A field is a type of integral domain where every non-zero element has a multiplicative inverse, meaning that for every element a in the field, there exists an element b such that a * b = 1.
#rings
#fields
#integraldomain