This paper continues to study the theory of the first paper by the author with symbols and notions appearing in this paper the same as in the first one if not specially stated. In order to state our main results, we first introduce some notions.An ideal a is called hypernilpotent, if there exists a finite number of positive intergers n_1, n_2,…, n_r such that a~(n1,n2,…,nr)=0.It is proved that a is hypernilpotent if and only if a is solvable, i.e. there exists an integer m ≥0 such that a~((m)) = 0.From the concept of hypernilpotent we can now define a radieal as follows: First, we can see easily that the union of all the hypernilpotent ideals of a non-associative and non-distributive ring R (briefly NAD-ring) may not be hypernilpotent. Furthermore, R may have hypernilpotent ideals. Let be the ideals of R such that is the union of all the hypernilpotent ideals of R In general, for every ordinal a which is not a limit ordinal, we define to be the ideal of R such that is the union of all the hypernilpotent ideals of R if a is a limit ordinal, we define In this way we obtain an ascending chain of ideals We may consider the smallest ordinal τ such that This ideal we shall call the radical of R.Definition 1: An NAD-ring R is called semi-simple,if the radicalThen we can state the following structure theorem.Theorem i: Let R be an NAD-ring with ascending chain condition (briefly a.e.e.) of ideals. Suppose that every prime ideal of R is maximal and R~2= R. ThenR is semi-simple if and only if R = R_1⊕R_2⊕... ⊕R_r,where R_i are non-nilpotent ideals which are simple rings.Theorem 2: Let R be a semi-simple NAD-ring with a. c.c. on ideals of R, then the following conditions are equivalent (i) R can be expressed uniquely as R= R_1⊕R_2⊕... ⊕R_r apart from the order of the R_i, where R_i are n on-nilpotent ideals which are simple rings.(ii) R~2= R, and every prime ideal is maximal.(iii) R~2= R, and every principle ideal (a) of R can be expressed uniquely as R_(i1)⊕R_(i2)⊕…⊕R_(is),where R_(ij) are non-nilpotent ideals, which are simple rings.If one of these prepositions holds, then so does the following.(i) every ideal a of R is principal and a~2 = a.(ii) every prime ideal p can be expressed uniquely as P_i = R_1⊕R_2⊕…⊕R_(i-1) ⊕R_(i+1)⊕…⊕R_r,i = i, 2…, r.(iii) the number of ideals of R is precisely(1r)+ (2r)+…+(rr)while the number of proper prime ideals of R is precisely r.Definition 2: The W-ascending chain conidtion (briefly w- a.c.c.) on ideals of R is said to hold in R, if for a given ascending chain on ideals a_1 a_2 …a_n … there exists a finite number of positive integers n_1,…,n-r and n such that W~(n1,…,nr)∩a_n= W~(n1,…,nr,)∩a_(n+1)=… where W = ∪ a_i.It is clear that every ring satisfying a. c.c. on ideals also satisftes w- a. c.c. on ideals.Theorem 3: The conclusions of theorems 1 and 2 are still valid, if NAD-ring satisfies w-a.c.c, on ideals and a.c.c. (i.e. descending chain condition) on ideals of R instead of a.c.c, on ideals of R.Definition 3: Let m be an ideal of R. An ideal a is called m-hypernilpotent,if there exists a finite number of positive integers n_,…,n_r such that a~(n1,…,nr)m. Otherwise a is called m-nonhypernilpotent.Theorem 4 : Let R be an NAD-ring with w- a. c. c. on ideals then we have the following results: (i) If m is an ideal, then there exists a semi-prime ideal p containing m such that every principal ideal (a) contained in p is m-hypernilpotent. Hence every semiprime ideal p containing m must cantain p.In this case we will say that p is m-semiprime.(ii) If a is an m-nonhypernilpotent ideal, then there exexists at least one anonhypernilpotent prime ideal p_a such that p_a m and that p_a≥p≥m implies p_a=p, where p is prime.In this case we will say that p_a is m-prime.(iii) every m-semiprime ideal p can be expressed as an intersection of m-prime ideals. If R satisfies a.c.c, on ideals, then p=p_1 ∩…∩ p_r where p_i are prime ideals, and then there exist two finite sets of positive integers n_1,…,m_t and k_(1,…,) k_s respectively such that p~(m1,…,m_t) m p~(k,…,k_s
