Editing Parity (mathematics) (section)

===Higher dimensions and more general classes of numbers===
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| Each of the white [[bishop (chess)|bishops]] is confined to squares of the same parity; the black [[knight (chess)|knight]] can only jump to squares of alternating parity.
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Integer coordinates of points in [[Euclidean space]]s of two or more dimensions also have a parity, usually defined as the parity of the sum of the coordinates. For instance, the [[Cubic crystal system|face-centered cubic lattice]] and its higher-dimensional generalizations (the ''D<sub>n</sub>'' [[Lattice (group)|lattices]]) consist of all of the integer points whose coordinates have an even sum.<ref>{{citation
 | last1 = Conway | first1 = J. H.
 | last2 = Sloane | first2 = N. J. A.
 | edition = 3rd
 | isbn = 978-0-387-98585-5
 | location = New York
 | mr = 1662447
 | page = 10
 | publisher = Springer-Verlag
 | series = Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]
 | title = Sphere packings, lattices and groups
 | url = https://books.google.com/books?id=upYwZ6cQumoC&pg=PA10
 | volume = 290
 | year = 1999}}.</ref> This feature also manifests itself in [[chess]], where the parity of a square is indicated by its color: [[Bishop (chess)|bishops]] are constrained to moving between squares of the same parity, whereas [[Knight (chess)|knights]] alternate parity between moves.<ref>{{citation|title=Chess Thinking: The Visual Dictionary of Chess Moves, Rules, Strategies and Concepts|first=Bruce|last=Pandolfini|author-link=Bruce Pandolfini|publisher=Simon and Schuster|year=1995|isbn=9780671795023|pages=273–274|url=https://books.google.com/books?id=S2gI_mExCOoC&pg=PA273}}.</ref> This form of parity was famously used to solve the [[mutilated chessboard problem]]: if two opposite corner squares are removed from a chessboard, then the remaining board cannot be covered by dominoes, because each domino covers one square of each parity and there are two more squares of one parity than of the other.<ref>{{citation|doi=10.2307/4146865|title=Tiling with dominoes| first=N. S.|last=Mendelsohn|journal=The College Mathematics Journal|volume=35|issue=2|year=2004| pages=115–120|jstor=4146865}}.</ref>

The [[Even and odd ordinals|parity of an ordinal number]] may be defined to be even if the number is a limit ordinal, or a limit ordinal plus a finite even number, and odd otherwise.<ref>{{citation|title=Real Analysis |last1=Bruckner|first1= Andrew M.| first2=Judith B.|last2=Bruckner|first3= Brian S.|last3=Thomson |year=1997 |isbn=978-0-13-458886-5 | page=37|publisher=ClassicalRealAnalysis.com | url=https://books.google.com/books?id=1WY6u0C_jEsC&pg=PA37}}.</ref>

Let ''R'' be a [[commutative ring]] and let ''I'' be an [[Ideal (ring theory)|ideal]] of ''R'' whose [[Index of a subgroup|index]] is 2. Elements of the [[coset]] <math>0+I</math> may be called '''even''', while elements of the coset <math>1+I</math> may be called '''odd'''.
As an example, let {{math|1=''R'' = '''Z'''<sub>(2)</sub>}} be the [[Localization of a ring|localization]] of '''Z''' at the [[prime ideal]] (2). Then an element of ''R'' is even or odd if and only if its numerator is so in '''Z'''.