For decades, the semiconductor industry has been steadily shrinking transistor sizes to fit more performance into a single silicon-based integrated chip. This technology has become the driving force for advances in education, transportation, and health, among others. However, transistor sizes are quickly approaching their physical limits (channel lengths are now only a few silicon atoms in length), and Moore's law will likely soon be brought to a stand-still despite many unique attempts to keep it going (FinFETs, high-k dielectrics, etc.). This technology must then be pushed further by exploring (almost) entirely new methodologies. Given the explosive growth of optical-based long-haul telecommunications, we look to apply the use of high-speed optics as a substitute to the digital model; where slow, lossy, and noisy metal interconnections act as a major bottleneck to performance. We combine the (nonlinear) optical Kerr effect with a single add-drop microring resonator to perform the fundamental AND-XOR logical operations of a half adder, by all-optical means. This process is also applied to subtraction, higher-order addition, and the realization of an all-optical arithmetic logic unit (ALU). The rings use hydrogenated amorphous silicon as a material with superior nonlinear properties to crystalline silicon, while still maintaining CMOS-compatibility and the many benefits that come with it (low cost, ease of fabrication, etc.). Our method allows for multi-gigabit-per-second data rates while maintaining simplicity and spatial minimalism in design for high-capacity manufacturing potential.