Rings
★★★★☆Undergraduate
📖Definition
A ring is a set with two binary operations (addition, multiplication). It's an abelian group under addition, multiplication is associative, and distributive laws hold.
📐Formulas
(R, +) is an abelian group
Abelian group under addition
(a · b) · c = a · (b · c)
Associativity of multiplication
a(b + c) = ab + ac, (a + b)c = ac + bc
Distributive laws
✏️Examples
예제 1
Show (ℤ, +, ×) is a ring.
예제 2
What are properties of matrix ring M₂(ℝ)?
⚡Applications
Number Theory
Ring of integers, algebraic number theory
Polynomials
Polynomial rings
Cryptography
Finite rings, lattice-based crypto
#환#대수구조#ring#algebraic structure