The key thing is that there is NO electric field within the perfect wire. So, there is no force acting on the electron, and thus no work done on it (while it's in the perfect wire).
This goes back to the definition of a perfect conductor (which the perfect wire is). Within a perfect conductor, there is no electric field. Instead, the charges (which have infinite mobility) rearrange themselves on the surfaces of the conductor in such a way as to perfectly cancel out any internal field.
So, the only fields in your circuit would be 1) in the battery, and 2) in the resistor.
I should also add that this is due to the approximation of the wire as 'perfect'. A real wire has some resistance, or equivalently, its charges don't perfectly reorder so as to perfectly cancel an internal field.
but Ohm's Law states that the voltage changes also because of the characteristic of the conductor (Resistance).
Actually, Ohm's law does not state that. Ohm's law is just a relationship between - in this case - 3 parameters: Resistance $R$, current $I$ and voltage $V$.
$$V=RI$$
But while it is the relationship between them, it does not state which ones are dependent and will change when another one changes. That depends on the situation.
In the case of a battery connected to a simple circuit, we know that the voltage is constant. Regardless of the resistance in the circuit, the voltage is always, say, 5 V across the two battery terminals. This is true when not connected (open-circuit, effectively infinite resistance), when connected (some specific value of resistance) and when short-circuited (effectively zero resistance).
The voltage is constant and doesn't depend on the resistance that happens to be in the way. So, when looking at Ohm's law, what is then changing, if not $V$? Mathematically, you are right that something else must change when $R$ changes. But it doesn't have to be $V$ - it can also be $I$. And it is.
Think of $V$ as the "pressure" that "pushes" on charges.
- In the open-circuit case, they are being pushed with 5 V, but they still can't move because there is no conducting path - infinite $R$ but zero $I$. Ohm's law obeyed.
- In the connected case, they are being pushed still with 5 V, and they flow with whichever current $I$ that fulfills Ohm's law. $R$ limits and alters $I$, not $V$.
- In the short-circuit case, they are still being pushed with 5 V and this time nothing stops them. They speed up and up and up. At any flow speed (current), the 5 V pushes them faster to gain higher speed, which continues forever (or until the heat generated melts the wire). $R$ is zero and $I$ infinite - Ohm's law obeyed.
In general, be careful when reading a mathematical formula. It only shows a relationship between parameters - it doesn't show which of them that are dependent and which that are fixed. That depends on the particular situation.
Best Answer
It's not zero.
This is a missaplication of ohms law.
Ohms law states the potential needed to maintain a constant current under a resistive force.
It is a steady state solution of a differential equation, when the applied electric force equals the resistive force
Ie, the condition that $\vec{a} = 0$
V=IR
When R=0, V=0
Why does this equation give zero? Because in the absence of resistive forces, what is the potential needed to maintain a CONSTANT current?
Clearly zero potential is needed to maintain a constant current as in the absence of resistive forces, the current will continue to move at a constant rate. Ie, zero potential is needed to maintain it.
This is all ohms law is saying, it is a steady state solution under the assumption there is no acceleration.
This is also why using ohms law at 0 resistance, we can say "I" Is anything, as when potential is zero, all currents satisfy the condition that the current is constant.
If a potential is applied to a superconducting wire, 0 potential is needed to maintain a constant current.
Does this mean the potential is zero? Obviously not. The potential will be whatever the applied potential is, and thus there obviously IS an electric field.
I apply a potential under zero resistance, the current is changing, and thus using ohms law in this way to say the potential is zero is false. This is only the case when I is constant, which would be in the absence of an applied potential difference, which only occurs, when I don't apply a potential difference.