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By Wedderburn's little theorem, every finite division ring is a field.
Finite division rings are commutative: Every finite division ring is commutative; equivalently, every finite division ring is a field.
This is the standard corollary of Wedderburn's little theorem, which asserts that any finite division ring has commutative multiplication. Thus a finite division ring is a field.
Wedderburn's little theorem: Every finite division ring is commutative. Equivalently, every finite division ring is a field.
A division ring (or skew field) is a unital ring with such that every nonzero element of is a unit (equivalently, is a group under multiplication, i.e. the group of units equals all nonzero elements).
A field is a commutative ring with such that every nonzero element is a unit (equivalently, every has a multiplicative inverse).