math-ring.html


* created: 2025-12-10T16:37
* modified: 2026-01-10T12:03

title

Ring

description

A group but with two binary opersions.

related notes

Ring

A ring (R, +, \cdot) is a non empty set, with two binary operations + and \cdot that fulfills the following conditions:

  1. (R, +) is a group
  2. (R, \cdot) is a semigroup
  3. The distributive rules hold for every a,b,c\in\mathbb{Z}_{n}:
    • a \cdot (b+c) \equiv (a\cdot b) + (a\cdot c) \pmod{n}
    • (a+b) \cdot c \equiv (a\cdot c) + (b\cdot c) \pmod{n}

Basically a group but with two binary operations, instead of one.

Field

Polynomial division over residue class fields

That's Polynomial Division with Rest. The main take away here would be, if we have two polynomials f, g \in K[x], then the following holds:

  1. If deg(f) < deg(g), then: q(x) = 0 and r(x) = f(x), i.e. the result is 0 \text{ Rest:} f(x)
  2. If deg(f) \geq deg(g), then do the polynomial division thingy