(MAJOR EDIT: I think I missed the associative part, is that more or less my only mistake?) I've got an un"well-formed" question, I've been staring at things like every ring is a module over itself, counting the number of sets and operations in various definitions of algebraic objects. I was looking at definition of algebra (over comm. ring), and thinking, "what's the point, it looks like a ring, what's the difference"? So what I just decided, and I'm not sure if this is best way to chop it up, is that: every algebra is a ring every ring is an algebra every noncommutative ring is an algebra over it's center, a comm. ring Is this correct? So algebras coincide with interest in noncommutative rings, and really just the noncommutative ring looked at as a module over it's center, the commutative ring part. If we look at the module over some units, then it is a vector space with a product for vectors, or an algebra over a field, or a noncommutative ring of a special type. Is this a decent/good/normal way to see these topics?