I recall reading in an abstract algebra text two years ago (when I had the pleasure to learn this beautiful subject) that there exists a division ring D that is not isomorphic to its opposite ring. However, the author noted that the construction of such a ring “would take us too far afield”. My question is: please give an example of a division ring D that is not isomorphic to its opposite ring.
Let me recall that a division ring is a ring A such that every non-zero element of A is a unit, i.e., has a multiplicative inverse in A. However, division rings are not required to be commutative. Let me also recall that if A is a ring, the opposite ring of A is the ring Aop that has the same underlying set as A, the same additive structure as A, but that the multiplication ∗ on Aop is defined by the rule a∗b=ba where “ba” denotes the product of b and a with respect to the multiplication in A.
It is not hard to find examples of rings which are not isomorphic to their opposite ring and it is trivial to find examples of noncommutative rings that are isomorphic to their opposite rings. (E.g., the n×n matrix ring over a field F (n>1) is isomorphic to its opposite ring via the transpose map A→AT where A is a matrix and AT denotes its transpose.) Of course, every commutative ring is isomorphic to its opposite ring via the identity map. However, this particular question appears to be difficult.
Let F be the center of the division ring D. Then D represents its class in the Brauer group Br(F). The opposite ring Dopp represents the inverse element. The reason why for example the quaternions are isomorphic to their opposite algebra is that the quaternions are an element of order 2 in Br(R), and hence equal to its own inverse in the Brauer group.
To get a division algebra that is not isomorphic to its opposite algebra we can use an element of order 3 in the Brauer group. One method for constructing those is to start with a (cyclic) Galois extension of number fields E/F such that [E:F]=3. Let σ∈Gal(E/F) be the generator. Let γ∈F be an element that cannot be written in the form γ=N(x), where N:E→F,x↦xσ(x)σ2(x) is the relative norm map. Consider the set of matrices
A theorem of A. Albert tells us that this forms a division algebra with center F, and its order in Br(F) is 3, so it will not be isomorphic to its opposite algebra. The theory is described for example in ch. 8 of Jacobson’s Basic Algebra II. The buzzword ‘cyclic division algebra’ should give you some hits.
For a concrete example consider the following. Let F=Q(√−3) and let
E=F(ζ9), with ζ9=e2πi/9. Then E/F is a cubic extension of cyclotomic fields, σ:ζ9↦ζ49. I claim that the element 2 does not belong the image of the norm map. This follows from the fact 2 is totally inert in the extension tower E/F/Q. Basically because GF(26) is the smallest finite field of characteristic 2 that contains a primitive ninth root of unity. Now, if 2=N(x) for some x∈E, then 2 must appear as a factor (with a positive coefficient) in the fractional ideal generated by x. But the norm map then multiplies that coefficient by 3, and as there were no other primes above 2, we cannot cancel that. Sorry, if this is too sketchy.
Anyway (see Jacobson again), the product of the γ elements modulo N(E∗) is the operation in the Brauer group Br(E/F)≤Br(F). Therefore the opposite algebra should correspond to the choice γ=1/2, (or to the choice γ=4, as 2⋅4=N(2)). So
and the choices γ=2 and γ=1/2 yield non-isomorphic division algebras, as their ratio =4 is not in the image of the norm map.
Edit (added more details here, because another answer links to this answer): In general the cyclic division algebra construction works much the same for any cyclic extension E/F. When [E:F]=n we get a set of n×n matrices with entries in E. The number γ appears in the lower diagonal part of the matrix. The condition for this to be a division algebra is that γk should not be a norm for any integer k,0<k<n. Obviously it suffices to check this for (maximal) proper divisors of n. In particular, if n is a prime, then it suffices to check that γ itself is not a norm.
Edit^2: Matt E's answer here linear algebra over a division ring vs. over a field gives a simpler cyclic division algebra of 3×3 matrices with entries in the real subfield of the seventh cyclotomic field.