It was pointed out in the comments that if we mean by "$\sum_i a_i$" just the $I$-tuple $(a_i: i \in I)$, then the family of such tuples can of course be made into a semi ring (e.g., by pointwise addition and multiplication, or Cauchy multiplication if you use the index set $\mathbb N$). But in this case, neither the notation/name "sum" makes sense to me, nor does question 2 make sense: $R^I$
is never empty for nonempty $R$.

So I will reinterpret the question: Which semirings can be extended to complete semirings? By "complete semiring" I mean a semiring $(S,+,\cdot,0,1)$ (with commutative addition) in which all sums $\sum_{i\in I} a_i$ (for all index sets $I$, or at least for all countable $I$) are defined and satisfy reasonable laws, in particular:

- infinite associativity and commutativity
- left distributivity: $c\cdot \sum_i a_i = \sum_i c\cdot a_i$, and similarly right distributivity; in particular, the sum of 0s is 0.
- finite sums agree with addition.

If you also impose the following (natural, I think) condition, called "d-completeness" or "discrete completeness":

- Whenever $\sum_{i=0}^n a_i = a_0$ for all $n\ge 0$, then also
$\sum_{n=0}^\infty a_i = a_0$

then it is easy to see that the following are equivalent.

The semiring $S$ is a subsemiring of a complete semiring.

$S$ can be partially ordered by $a \le b$ iff $\exists x\ a+x=b$.

For all $a,b,c$ in $S$, $a+b+c=a$ implies $a+b=a$. (Of course, as Darij Grinberg has pointed out, this is never true in rings.)

The point is that $a+(b+c)+(b+c)+\cdots = a+b + (c+b)+(c+b)+\cdots$. A few details can be found here.

Georg Karner's paper "On limits in complete semirings" (Semigroup Forum 45 (1992), no. 2, 148–165, MR1171841 (93h:16078)) has more information, and uses topological concepts rather than order-theoretic ones.

alwayshave $\hat{R}=0$. – Robert Kucharczyk Jan 13 '12 at 13:21