# units

• June 22nd 2011, 11:08 PM
alexandrabel90
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
• June 22nd 2011, 11:21 PM
topspin1617
Re: units
A field is a commutative ring such that every nonzero element is a unit.

$\mathbb{R},\mathbb{Q},\mathbb{C}$ 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.
• June 22nd 2011, 11:27 PM
Deveno
Re: units
that is almost correct. for a ring R, if every non-zero element is a unit, the proper term is division ring, as the multiplication may not be commutative.

the standard example is the quaternions $\mathbb{H} = \{a + bi + cj + dk: a,b,c,d \in \mathbb{R}\}$,

where $ij = k, jk = i, ki = j, ji = -k, kj = -i, ik = -j$ and $i^2 = j^2 = k^2 = -1$.

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 $\mathbb{Z}_6$.

note that (2)(3) = 0 (mod 6), so neither 2 nor 3 has an inverse (it would be like "dividing by 0").