In this abstract we denote the non-associative and non-distributive ring briefly by NAD-ring.Let {R_i} be a coUection of NAD-rings and denote L as the collection of all elements a = (a_1,a_2,…,a_i,…) where a_i ∈ R_i. As usual we can introduce into L the following additive operation and scalar multiplication: (a_1,…, a_i,…) + (a_1′,…, a_i′,…) = (a_1 + a_i′,…, a_i + a_i',…) aa = (aa_1,…,aa_i,…),aa= (a_1a,…, a_ia,…) where a∈. Let a be an additive subgroup of L and called a subgroup if and only if aa ∈a, aa ∈a for every a ∈and a ∈a. A-subgroup a is called a subsystem of L if and only if a can be represented by the usual complete direct sum of a collection {a_i}of systems a_i of R_i where a_i are the images of a into R_i under the natural homomorphism (a_1,…, a_i,…)→ (0,…, a_i, 0…) In this case we denote a=(a_1,…, a_i,…).Now we can introduce the multiplicative operation []into L.Let b=(b_1,…, b_i,…)∈L and a= (a_1,…, a_i,…) be a subsystem of L. We define the multiplication of b and a as follows [b, a] = ([b_1, a_1],…,[b_i,a_i],…) [a, b] = ([a_1, b_1],…,[a_i, b_i],…)It is easy to prove that L is an NAD-ring.Definition 1 : The above NAD-ring L is called the complete direct sum of the collection of NAD-rings R_i.Definition 2:A subsystem L is called a subring of L if and only if[L, L]L.A subring L is called an ideal if and only if [L, L]L, [L,L]L. A subring L ofL is called a subdirect sum of NAD-rings R_i if and only if the images of L under thenatural homomorphism (a_1,…,a_i,…) → ( 0,…,a_i, 0,…)are either zero or R_i forevery i.Definition 3: We say that an NAD-ring R can be imbedded into the complete direct sum L if and only if there exists a mapping which is a group isomorphism of the additive group R into the additive group L and the following conditions are satisfied: (i) ([b, a])[(b), (a)] for every b ∈ R and every subsystem a of R. (ii) The natural homomorphism (a_1,…, a_i,…)→ (0,…, a_i, 0,…) of(R) into R_i is an onto mapping for every i.It is clear that the above conceptions are not all coincided with the usual conceptions.Theorem 1: An NAD-ring R can be imbedded into the complete direct sum of a collection of NAD-rings R_i if and only if R contains a collection of ideals I_i such that ∩ I_i=0 and R/I_i R_i.Theorem 2: Let R be a semi-simple NAD-ring satisfying the W-maximal condition on ideals. Suppose that every prime ideal of R is maximal, then R can be imbedded into the complete direct sum of a collection of simple NAD-rings Ri. On the contrary, if R is isomorphic to a subdirect sum of a collection of simple rings R_i, then every prime ideal of R must be maximal.Theorem 3: Let R be a semi-simple NAD-ring satisfying the W-maximal condition on ideals of R and be isomorphic to a subdirect sum S of simple NAD-rings R_i, then every ideal a of R must be isomorphic to a subring (D_1,…, D_i,…)_s of S, where (D_1,…, D_i,…)_s denote the collection of the elements of (D_1,…, D_i,…) in S and D_i are either R_i or zero. Furthermore, every prime ideal of R must be isomorphic to (R_1, R_2,…, 0, R_a,…)_s in which only one zero appears, and every minimal ideal of R must be isomorphic to a R_i.Theorem 4: Let R be a semi-simple NAD-ring satisfying the maxiimal condition on ideals. Suppose that either R contains a proper ideal or every non-zero ideal a is a zero divisor, namely, there exists a non-zero ideal b such that [a, b] = 0, [b, a]=0. Then R is adirect sum of principal ideals.Definition 4: An NAD-ring R is called split if and only if for every pair ideals a and b such that there exists an ideal such that.Theorem 5: Let R be an NAD-ring satisfying the minimal condition on ideals, then R is split if and only if the intersection of maximal ideals of R is zero.Theorem 6: Let R be a split NAD-ring satisfying the minimal condition on ideals, and be the intersection of prime ideals of R. Suppose that R contains non-zero proper ideals. Then R = ⊕(a_1)⊕…⊕(a_n),= 0 and (a_i)~2= (a_i), where (a_i) are minimal ideals. i= 1,2,…,n. Let a be an ideal and then a=⊕(a_(i1))⊕…⊕(a_(is)) a