The following is taken from Hodge, Schlicker and Sundstrom Abstract Algebra an Inquiry Based Approach text
For each natural number $i,$ let $R_i$ be a ring. Define the infinite direct sum
$$\bigoplus_{i=1}^{\infty} R_i=R_1\oplus R_2 \oplus R_3 \oplus \cdots$$
to be the set of all sequences of the form
$$x=(x_1,x_2,x_3,\cdots),$$
where $x_i\in R_i$ for all $i,$ and $x_i=0_{R_i}$ for all but finitely many values of $i.$ To illustrate, let $R_i=Z_{i+1}$ for every natural number $i,$ and define
$$R=\bigoplus_{i=1}^{\infty} R_i=\bigoplus_{i=1}^{\infty}Z_{i+1}=Z_2\oplus Z_3\oplus Z_4 \oplus \cdots$$ Then
$x=([1]_2,[0]_3,[0]_4,[0]_5,[0]_6,[0]_7,[0]_8,[0]_9,\cdots)\in R_i$$
since $x_i$ is the zero element in $Z_{i+2}$ for all but $3$ values of $i.$ In contrast, $$y=([1]_2,[1]_3,[1]_4,[1]_5\cdots)\notin R_i,$$
since $y_i$ is nonzero for infinitely many values of $i.$
The characteristic of an infinite direct sum. Let $R$ be the infinite direct sum as defined above:
(a) Show that for every $x=(x_1,x_2,x_3,\cdots)\in R,$ there exists a positive integer $k$ such that
$kx=0_R=([0]_2,[0]_3,[0]_4,\ldots).$
(Hints: Recall that $x_i$ is nonzero for only finitely many values of $i.$).
(b) Show that, in spite of your conclusion in part (a), the characteristic of $R$ is zero.
Question: for (a) the positive integer $k$ is as follows, for any $x_i\in Z_{i+1}$, then $k={char}R={lcm}\{Z_{i+1\cdot i+2 \cdot i+3\cdots}\}={char}R={lcm}\{x_{i+1}\cdot x_{i+2}\cdot x_{i+3}\cdots\}$ where $i=1,2,\cdots n-1,$ since only finitely many $x_i$ are not zero. (I might not be accurate in my notation).
for part (b) I am not sure why the characteristic of $R$ is zero, other than if we take into account of all the values of $x,$ including the ones that are zero.
This is a matter of quantifiers. For any $x$ in the ring, there is an integer $k$ that annihilates it: $kx = 0$. However, this integer depends on the element of the ring. For positive characteristic, we have to have an integer that works for all elements in the ring.
And in this example, you're showing that no such integer exists. Explicitly, you're constructing a (countable) collection of ring elements that cannot be annihilated by any particular integer.
This is one of the (many) differences between finite and infinite sets. Every finite set has a least common multiple, but there are plenty of infinite sets that do not.