I’m looking into 32/64bit signed integers and their algebraic properties. I am quite sure that the the three operators +, – and * fulfill the distributive and associative property (integer division does not because of the information loss) even though we have some kind of information loss due to wrapping of positive <-> negative numbers.
What I’m looking for is some kind of academic publication (if it is not too trivial) that addresses this issue and provides some proof or counterexample.
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Integers modulo n (ℤ/nℤ) with addition and multiplication form a Commutative Ring, where multiplication distributes over addition. If n is prime, they even form a finite field.
I don’t see why subtracting n/2 from every number (to make it signed) would necessarily change that.
For a definitive answer, Mathematics.SE is probably the better audience.
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