Abstract Algebra 12.1: Definition of a Ring

In this video, we define an algebraic structure with two binary operations, called a ring. We also classify rings by certain properties that the rings may or may not have.

[bonbonpony]

00:54 So cannot rings be non-commutative? :/ What about quaternions then?
01:01 Oh really? So what if we try reusing the same binary operation on a subset (which is not necessarily a subgroup)? It might not be closed anymore, despite it being the same operation :q This makes me rather think that closure should be considered on its own ground, because it depends more on the set than on the operation.
01:28 In other words, it is a _semigroup_ under multiplication. Hmm… but what about the identity element? Shouldn't a ring have one? (i.e. be a _monoid_ under multiplication)
04:36 Well, for the usual algebraic structures at least, such as integers or matrices. But we shouldn't assume that it will _always_ be the case, should we? A similar assumption about multiplication being "always associative" once turned out to be flawed, so why are we possibly making the same mistake about addition? :q