When defining a principal ideal, e.g $I$
$I=aR=(ar:r \in R)$
(an ideal generated by a single element of the ring)
do we require the ring to have an identity element and if so, in what manner does this affect the structure of the principal ideal? More precisely, is the requirement to have an identity element necessary to allow the element that generates the principal ideal, to belong to it in a trivial fashion
($a \in I$)?
PS Hopefully this question is not a matter of semantics, involving the ages-old ring vs rng definition i.e. whether we consider the identity element as part of every ring or not.
Ok, I think I made some progress: If we do not assume that the ring $R$ has an identity element, but do assume that it is a commutative ring and every ideal of the ring is principal-that is, $R$ is a Principal Ideal Domain, we can show that $R$ indeed has an identity element as a result of this assumption.
Indeed, let $R$ be a ring where every ideal $I⊆R$ is a principal ideal. Then, since $R$ is an ideal of $R$, and since every ideal of $R$ is a principal ideal,there must exist an element $a\in R $ such that $R=<a>=aR$
Since $a\in R$, there must be an element $r \in R$, such that $a=ar$.
Let $b \in R$. It follows $b=ac$ for some $c \in R$.
We have $b=ac=(ar)c=(ac)r$.
So $r$ acts as an identity element for a random $b \in R$
We define $r$ to be the identity element of $R$.