Question

Example of division ring which is not field

Answer 1

A ring R in which each nonzero element is a unit is called a division ring or skew field. The following subset of M2(C) is called the set of quaternions, and provides the best known example of a division ring that is not a field.

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Answer 2

Answer:

Of course, all fields are division rings. The most familiar example of a division ring which is not a field is that of Hamilton's real quaternions H = {a0 + a1i + a2j + a3k : ai ∈ R}. ... Thus, H contains the field R as a subring which is contained in its center; this is referred to as an R-algebra.