Elements of N fold Ring

This article covers the theoretical proof’s of 1 Let A be a non-empty set and θ_1,θ_2 〖,θ〗_3,……,θ_(n+1) be binary operations on A . Then A=〖(A,θ〗_1,θ_2 〖,θ〗_3,……,θ_(n+1)) is said to be n fold ring if 〖(A,θ〗_1) is an abelian group 〖 (A,θ〗_2) is semi group , 〖 (A,θ〗_3) is semi group , …….〖 (A,θ〗_(n+1)) is semi group θ_2 is distributive over θ_1 , θ_3 is distributive over θ_1 , ……, θ_(n+1 )is distributive over θ_1 . 2 If A is a n-fold ring with zero element 0 Then for all a ,b ,c ϵ A 1) aQi0 = 0Qia = O, ∀ i = 2,3,----, n+1. 2) aQi(-b) = (-a)Qib = - (aQib), ∀ i =2,3,…… 3) (-a) Qi (-b) = aQib , ∀ i = 2131……., n+1 4) aQi (bQ1(-c)) = (aQib) Q1(aQi (-c)) , ∀ i = 2,3,……, n+1 5) (-1) Qi a = (-a) , ∀ i = 2,3,……., n+1. 6) (-1) Qi (-1) = 1 , ∀ I = 2,3,4,……, n+1. 3 A finite n fold integral domain is a n-fold field . 4 The set of units in a commutative n-fold ring with unity is a abelion group with respect to Q2 ,-------, Qn+1 . 5 Any nonempty subset S of a n-fold ring A = (A1 Q1, Q2, Q3,---------,Qn+1) Is called sub n-fold ring ; if S = (S, Q1,Q2,--------,Qn+1) is a n-fold ring . 6 A nonempty subset S of a n-fold ring A is a subring of A iff xQ1(-y) , xQiy ϵ S , ∀ x,y ϵ S & -