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Rngs#

Definition: Rng

A rng \((R, +, \cdot)\) (pronounced "rung") is a set equppied with two operations \(+: R \times R \to R\) und \(\cdot: R \times R \to R\) which have the following properties:

  1. Addition Properties
    • Associativity: \((a + b) + c = a + (b + c)\) for all \(a, b, c \in R\).
    • Commutativity: \(a + b = b + a\) for all \(a, b \in R\).
    • Identity: There exists some \(0_R \in R\) such that \(a + 0_R = a\) for all \(a \in R\).
    • Invertibility: For each \(a \in R\), there exists some \(a' \in R\) such that \(a + a' = 0_F\).
  2. Multiplication Properties
    • Associativity: \((a\cdot b) \cdot c = a \cdot (b \cdot c)\) for all \(a,b,c \in R\).
  3. Distributivity Properties
    • Addition over Multiplication: \(a \cdot (b + c) = a \cdot b + a \cdot c\) for all \(a,b,c \in R\)
    • Multiplication over Addition: \((a+b) \cdot c = a\cdot c + b \cdot c\) for all \(a,b,c \in R\)

Notation

When it is clear from context which ring we are talking about, we can write simply \(0\) and \(1\) for the additive and multiplicative identities, respectively.

Multiplication is often denoted as \(ab\) or \(a\times b\) instead of \(a\cdot b\).

Multiplying an element by itself \(n\) times is denoted as

\[ a^n = \underset{n \text{ times}}{\underbrace{a \times \cdots \times a}} \]