One fact about the Lipschitz integers (quaternions of the form $a + bi + cj + dk$ where $a, b, c, d$ are integers) is that the left-sided ideal generated by any element $Q$ has the same index in the additive group as does the right-sided ideal generated by $Q$.
I know this is a fact, but I don't know a simple abstract reason for it. But surely there must be some embarrassingly simple reason I'm overlooking! Thus, I ask at exactly what level of abstraction this sort of thing cleanly holds:
- Is this the case in any ring with an involution which preserves ring structure except for reversing the order of multiplication (like the conjugation operation on quaternions)?
- Failing that, is this the case in any ring of the form $R[i, j, k \; | \; i^2 = j^2 = k^2 = ijk = -1]$ where $R$ itself is a commutative ring?
- Failing all that... what is the right level of abstraction for this result? What is the short and sweet reason it is true?