Ring Theory
نویسنده
چکیده
♣ Rings . A ring is a non-empty set R with two binary operations ( + , · ) , called addition and multiplication, respectively satisfying : Axiom 1. Closure ( + ) : ∀x, y ∈ R , x + y ∈ R . Axiom 2. Commutative ( + ) : For every x, y ∈ R , x + y = y + x . Axiom 3. Associative ( + ) : ∀x, y, z ∈ R , x + (y + z) = (x + y) + z . Axiom 4. Neutral ( + ) : ∃ θ ∈ R , such that ∀x ∈ R, x + θ = θ + x = x . We denote θ by 0 . Axiom 5. Inverse ( + ) : ∀ a ∈ R , the equation a+ x = θ has a solution. Note that (R , + ) is an Abelian group. Axiom 6. Closure ( ·) : ∀x, y ∈ R , x · y ∈ R Axiom 7. Associative ( ·) : ∀x, y, z ∈ R, x · (y · z) = (x · y) · z . Axiom 8. Ditributive ( + , ·) :
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