Do C# and Java longs form a commutative ring? -

a ring standard mathematical structure describing objects can added , multiplied. c# , java signed longs obey properties of ring? example, multiplication long.min_value associative , distributive? assume in unchecked context.

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(definition copied wikipedia)

a ring set r equipped binary operations + , · satisfying following 3 sets of axioms, called ring axioms.

1. r abelian group under addition, meaning
   • (a + b) + c = + (b + c) a, b, c in r (+ associative).
   • a + b = b + a, b in r (+ commutative).
   • there element 0 in r such + 0 = in r (0 additive identity).
   • for each in r there exists −a in r such + (−a) = 0 (−a additive inverse of a).
2. r monoid under multiplication, meaning that:
   • (a ⋅ b) ⋅ c = ⋅ (b ⋅ c) a, b, c in r (⋅ associative).
   • there element 1 in r such ⋅ 1 = , 1 ⋅ = in r (1 multiplicative identity).
3. multiplication distributive respect addition:
   • a ⋅ (b + c) = (a ⋅ b) + (a ⋅ c) a, b, c in r (left distributivity).
   • (b + c) ⋅ = (b ⋅ a) + (c ⋅ a) a, b, c in r (right distributivity).

a commutative ring 1 multiplication commutative (meaning ⋅ b = b ⋅ a).

on platforms signed values overflow defined wrapping, signed , unsigned values behave in isomorphic fashion when fed +, -, *, &, ^, |, <<, , ~ operators, or when performing unchecked cast smaller type. behave differently when used relational operators, >>, %, , / operators, when casting or promoting larger types.

because unsigned values of given size behave ring, signed values well. note because of implicit promotion int, smaller types may not behave arithmetic rings because applying + , * values of such type may yield not value of type.