Confusion between principal ideal and ideal
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?
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$\begingroup$No. If $I$ is an ideal and $a\in I$ then every multiple of $a$ also belongs to $I$. But the converse is not true — there might be no one element $a\in I$ such that every element of $I$ is a multiple of $a$.
Consider for example the ring ${\mathbb Z}[x,y]$. Let $I$ be the set of polynomial $p(x,y)$ such that $p(0,0) = 0$. It is easy to verify that that $I$ is an ideal. However, there is no element $a\in I$ that divides every element in $I$. (In particular, there is no element $a\in I$ that divides both polynomials $x$ and $y$.)
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