Changes

==Alternative Definition==

==Alternative Definition==

Mathematicians often prefer the following definition,{{fact}} which, for integers, is equivalent to the one stated above: <blockquote>''A prime number is a non-unit (i.e., not 1 or -1 for the integers) which whenever it divides the product of two numbers will divide at least one of the factors.''</blockquote>

Beyond the integers, mathematicians are interested in other structures with addition and multiplication; they are called [[Ring_(mathematics)|ring]]s. When working in rings, mathematicians use a different definition for a "prime" element: <blockquote>''A prime element is a non-unit (i.e., not 1 or -1 for the integers) which whenever it divides the product of two numbers will divide at least one of the factors.''</blockquote>

Or in symbolic notation:

Or in symbolic notation:

<math>p \in \mathbb{Z}  \quad prime :\Leftrightarrow p \not\in \{-1, 1\} \wedge (\forall a,b \in \mathbb{Z}: p \vert ab \Rightarrow p \vert a \vee p \vert b) </math>

<math>p \in \mathbb{Z}  \quad prime :\Leftrightarrow p \not\in \{-1, 1\} \wedge (\forall a,b \in \mathbb{Z}: p \vert ab \Rightarrow p \vert a \vee p \vert b) </math>

The advantage of this formulation is that it can be generalized on other structures which allow for addition and multiplication, i.e., [[Ring_(mathematics)|ring]]s. The earlier definition involves the notion of [[irreducibility]].

For the integers, this definition is equivalent to the one stated above. However in general the two definitions are not equivalent. The earlier definition is still important in rings, and is given the name [[irreducibility]], since it literally states that the element cannot be broken into smaller pieces.

 

The preference for this alternate definition of "prime" is that this definition gives you unique [[factorization]] theorems analogous to the [[Fundamental Theorem of Arithmetic]]. For more information, consult the articles on [[Dedekind ring]]s and [[ring theory]], or an abstract algebra book<ref>'''Abstract Algebra: An Introduction''', Thomas Hungerford. Brooks Cole; 2 edition (1996)</ref>

==Unique Factorization==

==Unique Factorization==