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 & -