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.
EDIT:
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 $R$.
Best Answer
You may define a principal ideal of a commutative ring (or a principal right ideal of a ring) that way, but it is not standard. The problem is that you have no guarantee $a$ is in $aR$ if that is your definition. This is what having an identity does for you, and it is supposed to be an expectation of and ideal "generated by an element."
Take for example the rng $2\Bbb Z$ and the principal ideal generated by 2 via your definition: $2\notin 2(2\Bbb Z)=4\Bbb Z$.
A ring whose ideals are all principal is called just a principal ideal ring.
I'm not sure what you want here. You are defining is structure explicitly. For rings not necessarily having identity, the normal definition of the principal right ideal is $a\Bbb Z +aR$, and the normal definition of the two sided principal ideal is the set of elements of the form $na+ \sum r_ias_i $ where n is an integer and the sum is finite, and $r_i,s_i$ are ring elements.
In the standard sense, $R=2\Bbb Z$ is a principal ideal domain without identity, but in your sense it is not (the entire ring can't be of the form $aR$) Requiring every ideal to be of that form is a very strong condition.