Editing Quotient ring (section)

=== Quaternions and variations ===
Suppose <math>X</math> and <math>Y</math> are two non-commuting [[indeterminate (variable)|indeterminate]]s and form the [[free algebra]] {{tmath|1= \mathbb{R} \langle X, Y \rangle }}.  Then Hamilton's [[quaternion]]s of 1843 can be cast as:
<math display="block">\mathbb{R} \langle X, Y \rangle / ( X^2 + 1,\, Y^2 + 1,\, XY + YX )</math>

If <math>Y^2 - 1</math> is substituted for {{tmath|1= Y^2 + 1 }}, then one obtains the ring of [[split-quaternion]]s. The [[anti-commutative property]] <math>YX = -XY</math> implies that <math>XY</math> has as its square:
<math display="block">(XY) (XY) = X (YX) Y = -X (XY) Y = -(XX) (YY) = -(-1)(+1) = +1</math>

Substituting minus for plus in ''both'' the quadratic binomials also results in split-quaternions.

The three types of [[biquaternion]]s can also be written as quotients by use of the free algebra with three indeterminates <math>\mathbb{R} \langle X, Y, Z \rangle</math> and constructing appropriate ideals.