One of the hardest and most fundamental problems in topology is computing homotopy groups of spheres. One can simplify it a little by restricting to the stable range. This reveals a lot more structure: stable homotopy groups form a graded ring. We will argue that it is useful to make it into a « homotopical ring », where we do not phrase relations (like commutativity) in terms of equality, but rather homotopy. We use this point of view to showcase a fundamental difference between algebra and homotopical algebra. In homotopical algebra, ring quotients usually do not exist. This is an expository talk.