Recent technological advances in integrated photonics have spurred on the study of topological phenomena in engineered bosonic systems. Indeed, the controllability of silicon ring-resonator arrays has opened up new perspectives for building lattices for photons with topologically nontrivial bands and integrating them into photonic devices for practical applications. Here, we push these developments even further by exploiting the different modes of a silicon ring resonator as an extra dimension for photons. Tunneling along this synthetic dimension is implemented via an external time-dependent modulation that allows for the generation of engineered gauge fields. We show how this approach can be used to generate a variety of exciting topological phenomena in integrated photonics, ranging from a topologically-robust optical isolator in a spatially one-dimensional (1D) ring-resonator chain to a driven-dissipative analog of the 4D quantum Hall effect in a spatially 3D resonator lattice. Our proposal paves the way towards the use of topological effects in the design of novel photonic lattices supporting many frequency channels and displaying higher connectivities.