Editing Quotient ring (section)

=== Variations of complex planes ===
The quotients {{tmath|1= \mathbb{R} [X] / (X) }}, {{tmath|1= \mathbb{R} [X] / (X + 1) }}, and <math>\mathbb{R} [X] / (X - 1)</math> are all isomorphic to <math>\mathbb{R}</math> and gain little interest at first. But note that <math>\mathbb{R} [X] / (X^2)</math> is called the [[dual number]] plane in geometric algebra. It consists only of linear binomials as "remainders" after reducing an element of <math>\mathbb{R} [X]</math> by {{tmath|1= X^2 }}.  This variation of a complex plane arises as a [[subalgebra]] whenever the algebra contains a [[real line]] and a [[nilpotent]].

Furthermore, the ring quotient <math>\mathbb{R} [X] / (X^2 - 1)</math> does split into <math>\mathbb{R} [X] / (X + 1)</math> and {{tmath|1= \mathbb{R} [X] / (X - 1) }}, so this ring is often viewed as the [[Direct sum of algebras|direct sum]] {{tmath|1= \mathbb{R} \oplus \mathbb{R} }}.
Nevertheless, a variation on complex numbers <math>z = x + yj</math> is suggested by <math>j</math> as a root of {{tmath|1= X^2 - 1 = 0 }}, compared to <math>i</math> as root of {{tmath|1= X^2 + 1 = 0 }}. This plane of [[split-complex number]]s normalizes the direct sum <math>\mathbb{R} \oplus \mathbb{R}</math> by providing a basis <math>\left\lbrace 1, j \right\rbrace</math> for 2-space where the identity of the algebra is at unit distance from the zero. With this basis a [[unit hyperbola]] may be compared to the [[unit circle]] of the [[complex plane|ordinary complex plane]].