Changes

<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>

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.

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 an abstract algebra book<ref>'''Abstract Algebra: An Introduction''', Thomas Hungerford. Brooks Cole; 2 edition (1996)</ref>

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 an abstract algebra book<ref>'''Abstract Algebra: An Introduction''', Thomas Hungerford. Brooks Cole; 2 edition (1996)</ref>