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algebrat
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(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:
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?
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?
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