Let $K\subset \mathbb Z$ be a subset of integers and let \(a_k, b_k, c\in R\) be any elements of a given unit ring: \((R, +, \cdot)\). Then the following rules for sum manipulation hold:
\[ \begin{array}{ccl} \sum_{k\in K}c\cdot a_k=c\cdot\sum_{k\in K}a_k&\quad\quad\quad&\text{distributive law},\\ \sum_{k\in K}(a_k + b_k)=\sum_{k\in K}a_k + \sum_{k\in K}b_k &\quad\quad\quad&\text{associative law},\\ \sum_{k\in K}a_k=\sum_{\pi(k)\in K}a_k&\quad\quad\quad&\text{commutative law}. \end{array} \] where \(\pi(k)\) is any permutation of the set \(K\).
Proofs: 1
