817 - Algebra

Rings

Definition: A ring is a set \(R\) with two binary operation \( +\) and \(\cdot\) satisfying:

1. \( (R,+)\) is an abelian group (with identity 0)
2. \( (R,\cdot) \) is a semigroup
3. \(\cdot \) is distributive over \( + \) (on both sides)

Named Rings

Euclidian Domains (EDs)

A Euclidean Domain (ED) is a domain \(R \) together with a function \(N: R \to \mathbb{Z}_{\geq 0} \) such that \(N(0)=0 \) and the following property holds: for any two elements \( a,b \in R \) with \(b\neq 0 \), there are elements \( q,r \in R\) such that \( a=qb+r \) and either \(r=0 \) or \(N(r)<N(b) \)