the ring of numbers

Elementary number theory is largely about the ring of integers, denoted by the symbol $\integer$ . The integers are an example of an algebraic structure called an integral domain. This means that $\integer$ satisfies the following axioms:

(a) $\integer$ has operations + (addition) and $\cdot$ (multiplication). It is closed under these operations, in that if $m, n \in \integer$ , then $m + n \in \integer$ and $m\cdot n \in \integer$ .

(b) Addition is associative: If $m, n, p \in \integer$ , then

$$m + (n + p) = (m + n) + p.$$

(c) There is an additive identity $0 \in \integer$ : For all $n \in \integer$ ,

$$n + 0 = n \quad\hbox{and}\quad 0 + n = n.$$

(d) Every element has an additive inverse: If $n \in \integer$ , there is an element $-n \in \integer$ such that

$$n + (-n) = 0 \quad\hbox{and}\quad (-n) + n = 0.$$

(e) Addition is commutative: If $m, n \in \integer$ , then

$$m + n = n + m.$$

(f) Multiplication is associative: If $m, n, p \in \integer$ , then

$$m\cdot (n\cdot p) = (m\cdot n)\cdot p.$$

(g) There is an multiplicative identity $1 \in \integer$ : For all $n \in \integer$ ,

$$n\cdot 1 = n \quad\hbox{and}\quad 1\cdot n = n.$$

(h) Multiplication is commutative: If $m, n \in \integer$ , then

$$m\cdot n = n\cdot m.$$

(i) The Distributive Laws hold: If $m, n, p \in \integer$ , then

$$m\cdot (n + p) = m\cdot n + m\cdot p \quad\hbox{and}\quad (m + n)\cdot p = m\cdot p + n\cdot p.$$

(j) There are no zero divisors: If $m, n \in \integer$ and $m\cdot n = 0$ , then either $m = 0$ or $n = 0$ .

 

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