
units
i read somewhere that a set of elements is a field iff all the elements are units except 0... but i thought that a ring R is a field iff they are commutative?
from how i see, elements that are commutative might not be units.
may i know what is wrong with my understanding?
thanks

Re: units
A field is a commutative ring such that every nonzero element is a unit.
$\displaystyle \mathbb{R},\mathbb{Q},\mathbb{C}$ are all fields. That all three are commutative rings is a fact from elementary algebra. Also, in all three rings, we are able to divide by any nonzero element.

Re: units
that is almost correct. for a ring R, if every nonzero element is a unit, the proper term is division ring, as the multiplication may not be commutative.
the standard example is the quaternions $\displaystyle \mathbb{H} = \{a + bi + cj + dk: a,b,c,d \in \mathbb{R}\}$,
where $\displaystyle ij = k, jk = i, ki = j, ji = k, kj = i, ik = j$ and $\displaystyle i^2 = j^2 = k^2 = 1$.
however, it is a theorem (due to Wedderburn) that every finite division ring is indeed commutative: http://math.colgate.edu/math320/dlan...Wedderburn.pdf
for an example of a finite ring that is NOT a field: consider $\displaystyle \mathbb{Z}_6$.
note that (2)(3) = 0 (mod 6), so neither 2 nor 3 has an inverse (it would be like "dividing by 0").