Suppose current runs through a resistor from left to right, and we define the left-to-right direction as positive.
Then from left to right, the potential decreases. So $V,$ the voltage across the resistor (defined as the electric potential at the right of the resistor minus the electric potential at the left of the resistor), is negative, while $I,$ the current through the resistor, is positive. So it would seem $V = -IR$ is the correct relationship, but I don't recall ever seeing Ohm's law written this way. Why not?
(For example, in the introductory textbook I have on hand, Ohm's law is derived using $E = V/\Delta L,$ even though, in the previous chapter, the book writes $E_x = -\Delta V_x/\Delta x.$ [1] The intermediate book I have on hand argues that $V$ and $I$ must be proportional, saying "$V$ is the line integral of $\mathbf{E}$ on a path through the conductor…", but it would seem more appropriate to say "$V$ is the negative of the line integral of $\mathbf{E}$…" The book defines $V$ as "the difference in potential between those terminals" without specifying the direction. [2])
[1] Physics, Resnick, Halliday, and Krane, 5ed. equations 29-12 and 28-37
[2] Electricity and Magnetism, Purcell and Morin, 3ed. text before equation 4.12
Best Answer
In the comments, Hal Hollis pointed to the "passive sign convention".
The passive sign convention is defined like this:
So it is simply taken as a definition that the positive direction for current is the opposite direction as the positive direction for voltage. With this convention, $V=IR$ is the correct equation.