Cracking the neural code is one of the central challenges of neuroscience. Neural codes allow the brain to represent, process, and store information about the outside world. Unlike other types of codes, they must also reflect relationships between stimuli, such as proximity between locations in an environment. In this talk, I will introduce neural rings and ideals, which are analogous to Stanley-Reisner rings and ideals, as valuable tools for understanding the structure of neural codes. More generally, we will see how techniques from commutative algebra and algebraic topology are useful for determining whether or not a neural code is convex.