Must PID contain 1? Must PID contain 1? My concern arises when i consider the gcd of say a and b in the PID. Since it is a PID, it is generated by one element say k. k obviously $\in (k)$. However, if PID does not contain 1, then i can't write $k = k \cdot 1$, can 1? So is it true that then k must be equal to $k = k \cdot q$ for some $q \in$ PID. I find this kind of weird. Am I misunderstanding anything?
 A: In most contexts involving domains, an identity is assumed to exist.
In principle you can define a PID without identity, but it is not likely you will encounter it in practice or text.
The main issue is that principal ideals are no longer simply multiples of their generator. In fact, $(x)=\{nx+ rx\mid n \in \Bbb Z, r\in R\}$, where $nx$ is special notation for "x added to itself n times" with the obvious accommodations for negative integers and zero.
You are completely correct in saying that $k\cdot 1$  does not represent a product of two ring elements, because the second symbol is it defined. Yet the ideal must contain $k$, by definition.
A: From wikipedia:
"Integral domain" is defined almost universally as above, but there is some variation. This article follows the convention that rings have a 1, but some authors who do not follow this also do not require integral domains to have a 1

So authors who ask rings to have a $1$ ask their domains to have a $1$, but authors who don't ask for their rings to have a $1$ don't ask for their domains to have a $1$ and don't ask it of  their PID's either.
I personally would suggest that you ask rings to have an identity and learn the theory there, I have heard from my proffesors that you can later study the subject without asking for rings to have $1$ without much difficulty (Authors who ask that their rings have a $1$ call the rings without $1$ Rngs since they lack the $i$ for identity)
A: A PID is in particular a domain, which is required to contain a $1$. 
A: May has or not. For Z: integers, we can say 2Z is a PID but it does not have 1.
PID means: every ideal of it is generated by only one element. And it does not have zero divisor. Domain part says that.
R is a domain <=> every a,b element of R, if ab=0 => a=0 or b=0
