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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
A field is a commutative ring such that every nonzero element is a unit.
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.
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,
whereand
.
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.
note that (2)(3) = 0 (mod 6), so neither 2 nor 3 has an inverse (it would be like "dividing by 0").