Artin defines an ideal $I$ as :
- $I$ is a subgroup of $R^+$
- If $a \in I$ and $r \in R$ , then $ra \in I$
And Principal Ideal is defined as
"In any ring, the set of multiples of a particular element $a$ , forms an ideal called a principal ideal generated by $a$"
My question is:
If the set of multiples of a particular element is called principal ideal then that automatically is one of the properties of an ideal (Prop 2), then is every ideal a principal ideal?