5 ms·
unital -> there is a multiplicative identity 1 such that 1.a = a for all a. normed -> there is a norm. A way of saying how big an element is. In the reals |x|
by 77pt77 8y ago
unital -> there is a multiplicative identity 1 such that 1.a = a for all a.
normed -> there is a norm. A way of saying how big an element is. In the reals |x| - x if x is positive and -x if x is negative. In the complex numbers |a+bi| = (a^2 + b^2)^(1/2)
Division -> Division (except possibly by zero) is always possible. That means for a and b not zero there exists c such that a=cb (c is a divided by b)
I could be wrong about these things since I'm quite rusty
- ducttapecrown 8y agoI think division is normally framed rather as the existence of multiplicative inverses for all non-zero elements of the ring. That is, R is a division ring if a) there exists a 1 in R b) for all x in R, there exists y in R such that x * y = 1. I'm pretty sure these are equivalent. Proof: (==>) Let a = 1, and b in R be arbitrary. Then there exists a c such that b * c = 1. So inverses exist! (<==) Let a and b in R be arbitrary. Then a = a * b^-1 * b, so c = a * b^-1. QED.