Marginal rate of substitution notation:
It is implicit in the interpretation: 

Mas-Collel: the amount that must be given (+)  to compensate for a reduction (-). 

Reny: The rate at which good j  can be exchanged (+ &amp; -) for good i. 

The derivation from total differentiation only requires the utility to be constant, so the derivative must be negative to express that if the quantity of good $i$ increases, the quantity of good $j$ should decrease.

Another reason why some authors omit the negative sign is that in equilibrium  $|MRS_{ij}|=|\frac{P_i}{P_j}|$ So either you use the derivation from total differentiation and also the negative ratio of prices, or leave both sides as positive and focus on interpretation. Most people prefer the latter.