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This article covers the theoretical proof s of 1 Let A be a non-empty set and _1, _2 , _3, , _(n+1) be binary operations on A . Then A= (A, _1, _2 , _3, , _(n+1)) is said to be n fold ring if (A, _1) is an abelian group (A, _2) is semi group , (A, _3) is semi group , . (A, _(n+1)) is semi group _2 is distributive over _1 , _3 is distributive over _1 , , _(n+1 )is distributive over _1 . 2 If A is a n-fold ring with zero element 0 Then for all a ,b ,c 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 S , x,y S & -

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Students in Mathematics , has a experience of arround 20 years teaching in Tumsar(sns) , Banaras , Gondia (miet) , Nagpur ( inc ) , Warora ( anc) India ; Published arround 65+ international repute journal research articles. Established Uttakarsh Mathematical Group Warora 442907 India.

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