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Homedivision ring
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division ring
A division ring is a ring $D$ with identity such that

$1\neq 0$

For all nonzero $a\in D$, there exists $b\in D$ with $a\cdot b=b\cdot a=1$
Every field is a commutative division ring. The Hamiltonian quaternions are an example of a division ring which is not a field.
Synonym:
skew field
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Nonzero have inverse by Henry ✘
Zero by Henry ✓
Linking shenanigans by waj ✘
link is wrong by Mathprof ✓
comment by Mathprof ✓
Zero by Henry ✓
Linking shenanigans by waj ✘
link is wrong by Mathprof ✓
comment by Mathprof ✓