Suppose we have an Inertial frame S, all the clocks in this frame are synchronized. Now suppose, two seperate events occur at two different place A and B in that reference frame. Now, the events are said to be simultaneous if the clocks at the respective places registers the same time for each event(i.e two observers at the respective places read the same time of occurrence for each event). Now, suppose we have an observer at some different place C & the distance from A to C & that from B to C are not equal. Now my question is that: will the observer at C also claim about these two events to be simultaneous? If "No" is the answer, then we have observers in one single inertial frame, disagreeing about the simultinity of two events. See, there is no relative velocity between the observers, all they are in a single inertial reference frame at different places(I'm strictly talking about a single inertial frame. No relative velocity between the observers, nothing!) How this can be possible that the observers in a single reference frame are not in agreement about the simultinity of two seperate events though all the clocks in this frame are synchronized from the beginning?
A reference frame assigns a time interval and a spatial interval to any pair of events. That assignment is what a reference frame is.
So by definition, if you and I share a reference frame, we will assign the same time interval to any pair of events. In particular, we will either both assign the time interval zero (in which case we will agree that the events are simultaneous) or we will both assign some other time interval (in which case we willl agree that they are not).
You're right! You can verify this easily by using the Lorentz transformations! The two observers A and B are in the same inertial frame so there is no need for a Lorentz transformation! On the other hand, the observer C is in an other inertial frame an thus is moving with some uniform speed relative to the other observers A and B. Therefore to go from one inertial frame to the other we must make a Lorentz transformation. Hence, to compare the observation of C with the observation of A and B, we first apply a Lorentz transformation to the observations. By doing so we will have that the time the light travelled from one event will have been shorter than it did for the other event! So note the importance of the speed of light which is not infinite!!
One could imagine a theory in which the meaning of "the same reference frame" is not clear between two different points in space, and in fact that is the subject of our modern understanding of gravitation, variously called either the general theory of relativity the theory of general relativity. In this theory, two people who believe that they are maintaining a constant distance between each other might fundamentally fall out of agreement on what clocks are in sync, if they are even ever in agreement in the first place.
However, if we are not including gravity, then general relativity takes the "special" form that we call either the special theory of relativity or the theory of special relativity, rather than being fully general. In this special case of the theory, if two observers are at rest relative to each other, they will agree on whether any two events are simultaneous and therefore they will agree on whether any two clocks are synchronized. It is only when one accelerates relative to the other that they find that the clocks ahead of them (in the direction they're accelerating) are "ticking slightly fast" and the clocks behind them are "ticking slightly slow" and therefore clocks fall out of sync.
The special case was in fact discovered a long time before the general theory, which requires much more advanced mathematics. It turns out that the general theory can be viewed as a "patchwork" of the special case across the more general spacetime, with the special case happening whenever a reference frame is in "free-fall" with respect to the local effects of gravity. In this respect the disagreement that one sees in the general theory is just another form of the disagreement that one sees whenever one is accelerating; Nature "wants" us to be in free-fall and we stubbornly refuse to pass through the floor and instead find ourselves accelerating in opposition to that free-fall to remain in one place; this inherently means that all of the clocks "above" us are ticking slightly faster and all of the clocks "below" us are ticking slightly slower.