What is the form of an element in the intersection of principal left ideals? let $R$ be a ring and $I$ be an index set. Suppose $x \in \bigcap Ra_{i}$ where $a_{i}\in R$ for every $i \in I$.
Does $x= \sum_{i=1}^{n} r_{i}a_{i}$ where $r_{i} \in R$ for every $i \in I$? i.e.  is $x$ is a finite linear combination of the $a_{i}$?
 A: The answer to your second question is yes, an infinite sum is undefined as an element of your ring (unless you can assign it a value like you do with convergent series in $\mathbb{R}$).
The answer to your first question is no. Consider the ideals $(2)$ and $(3)$ in $\mathbb{Z}$, then $1= (-1)\cdot 2 + (1)\cdot 3$ is a finite linear combination of $2$ and $3$ but $1\not \in (2)$ and $1\not\in (3)$.
From the equation $1=(-1)\cdot 2+ (1)\cdot 3$ you can write any element in $\mathbb{Z}$ as a combination of $2$ and $3$, but the set of such linear combinations is not the intersection in general.
In general the product of ideals is contained in the intersection, but it need not be equal always. For example $(2)\cap (3)= (6)=(2)\cdot(3)$. However $(2)\cap (4)= (4)$ and $(2)\cdot(4)=(8)$.
A: 
Suppose $x \in \bigcap Ra_{i}$ where $a_{i}\in R$ for every $i \in I$.
Does $x= \sum_{i=1}^{n} r_{i}a_{i}$ where $r_{i} \in R$ for every $i \in I$? i.e.  is $x$ is a finite linear combination of the $a_{i}$?

Mostly yes, but a little bit no. Mostly yes, because for all $i\in I$, there exists $r_i$ such that $x=r_ia_i$, and that is technically a linear combination.  I say "a little bit no" because phrasing it in terms of linear combinations is confusing and unnecessary.
The elements of the intersection don't have a single description in terms of all the generators, they have multiple descriptions using each principal ideal generator separately.
The description is more appropriate for $\sum_{i\in I} Ra_i$, where every element indeed is of the form you proposed.
The sum and intersection are diametrically different creatures.  Considering their place in the poset of left ideals of $R$, the sum is an upper bound (in category theory terms it is a type of colimit) and the intersection is a lower bound (in category theory, a type of limit).
