Well, here’s a useful Wikipedia article about Algebra that somewhat relates to the previous thread.
Note that what most people learn in high school is only considered “Elementary algebra”. I remember as an undergraduate in the Math department being surprised to see that some of the graduate level mathematics classes used textbooks like “Introduction to Algebra” or something like that. The algebra you learn in high school is really only a specific case of an algebra, but the study of algebra involves a lot more interesting things.
By the way, note how in the examples of Algebras listed in the Wikipedia article how the examples given are specifically stated that multiplication EXCLUDES Zero. This is because as we discussed before, for none of those examples is division by zero defined within that algebraic system. In fact, in most “standard” algebraic systems, the term zero is given for an element which is an additive identity, which often ends up meaning that it will end up being an exception for multiplication, if addition and multiplication correspond at all to their intuitive meanings. There are exceptions of course, but they are usually more bizarre (such as systems where there is more than one zero).