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The image of {{mvar|x}} in the quotient is the "imaginary" unit {{mvar|j}}. With this description, it is clear that the split-complex numbers form a [[commutative algebra (structure)|commutative algebra]] over the real numbers. The algebra is ''not'' a [[field (mathematics)|field]] since the null elements are not invertible. All of the nonzero null elements are [[zero divisor]]s.
The split-complex numbers are [[algebra isomorphism|isomorphic as an algebra]] to the [[direct product]] of {{math|'''R'''}} by itself. It is thus a [[reduced ring|reduced]] [[Artinian ring]].
Since addition and multiplication are continuous operations with respect to the usual topology of the plane, the split-complex numbers form a [[topological ring]].
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