Discussing the differentia of Medical- Diagnostic Expert System(MDES), this paper gives an Associative Memory Neural Network(AMNN) appropriative to the inference engine of MDES.

Hence, an associative interpolation algorithm is proposed in this paper for improving the learning accuracy of CMAC. Meanwhile, a simulation experiment is described.

Su Yucai and Zhao Kaiming introduced the Weyl type algebra A[D] and proved that A[D], as an associative algebra, is simple if and only if A is D-simple and κ1 [D] acts faithfully on A.

Implications for the polymerization mechanism are discussed; the process follows an associative interchange (Ia) pathway.

It is shown that any neutral polyverbal subgroup W is generated as a subgroup by the set of its fully neutral polywords, and a necessary and sufficient condition is given for an associative neutral polyverbal operation to be verbal.

It is proved that ifR is an associative algebra with identity element over an infinite fieldF, then the algebraR(-) is nilpotent of lengthc if and only if the semigroupM(R) (orA(R)) is nilpotent of lengthc (in the sense of A.

Let R be an associative ring with unit, let S be a semigroup with zero, and let RS be a contracted semigroup ring.

Infinite Independent Systems of Identities of an Associative Algebra over an Infinite Field of Characteristic Two

In this paper we first give a definition for a module over a non-assoeiative and non-distributive ring (briefly NAD-ring). It can easily be seen that the notion of module over NAD-ring contains the notion of module over an associative ring in usual sense. However, we have constructed an example which shows that the module over NAD-ring can not be always an usuat module over an associative ring. Using this notion of module over an NAD-ring we can introduce the notion of primitive NAD-ring. Then we...

In this paper we first give a definition for a module over a non-assoeiative and non-distributive ring (briefly NAD-ring). It can easily be seen that the notion of module over NAD-ring contains the notion of module over an associative ring in usual sense. However, we have constructed an example which shows that the module over NAD-ring can not be always an usuat module over an associative ring. Using this notion of module over an NAD-ring we can introduce the notion of primitive NAD-ring. Then we define Jacobson-radical of NAD-ring. It can be proved that the Jacobson-radical J of an NAD-ring R can be expressed as the intersection of all maximal normal regular right ideals of R and that the Jacobson-radical of the residue NAD-ring R/J is O. Therefore we can introduce the notion of semi-simple NAD-ring. In this paper we give a method of characterizing the Jacobson-radical. Using this characterization we obtain the following main results:Structure theorem. Let R be a semi-simple non-associative and non-distrlbutivering, satisfying the minimal condition for right ideals of R, then(ⅰ) R has a finite number of dimple ideals (a_1), (a_2),…, (a_n) of R such that R is a direct sam of these.(ⅱ) Every ideal A of R is a direct sam of (a_i_1), (a_i_2),…, (a_i_5) which are some of (a_1), (a_2),…, (a_n). If the ideal A of R is considered as a ring, then every (right) ideal of A is also a (right) ideal of R. Moreover, R has exactly 2~n number of non-zero ideals of R.(ⅲ) Every right ideal of R is a direct sum of a finite number of minimal right ideals of R.(ⅳ) Every prime ideal of R must be maximal and the number of prime ideals of R is n exactly, n being the number of (a_1),…, (a_n) given by (ⅰ). Moreover every prime ideal p_i is the form p_i=(a_1)+…+(a_(i-1)) + (a_(i+1))+…+ (a_n).In particular, if our NAD-rings are the usual associative rings then our results are consistent with the well-known results.

In the literature, only the tensor product SL of a right R module S and a left R module L is defined and considered, where R is a ring. The theory of such tensor products is very important in the theory of rings, modules, categories and homology. However, such a tensor product is not a ring module, neither a left R module, nor a right R module. In this paper, the tensor product L_1L_2 of two left ring modules is defined, where L_i is a left R_i module, i=1, 2, R_1 and R_2 are K-rings, may not equal. Thus, if...

In the literature, only the tensor product SL of a right R module S and a left R module L is defined and considered, where R is a ring. The theory of such tensor products is very important in the theory of rings, modules, categories and homology. However, such a tensor product is not a ring module, neither a left R module, nor a right R module. In this paper, the tensor product L_1L_2 of two left ring modules is defined, where L_i is a left R_i module, i=1, 2, R_1 and R_2 are K-rings, may not equal. Thus, if is a category of a class of lelt ring madules L_μ, associated with morphisms H (L_μ, L_λ), L_μ, being left R_μ module and if L' is a left R' module, then—L' is a functor of . A part of the results obtained are the following: Let K be a commutative ring with unity. In §2, we considered the tensor product of some well-known K rings (wrt. K). Some of them are. If R_1 and R_2 are division rings, K is a field, and if a∈R_1, β∈R_2, such that they have the same minimum polynomial, over K, then R_1R_2 can't be a division ring. Let R_1 and R_2 be simble rings over a field K, with centrum ∑_1 and ∑_2 resp. Let F=K(∑_1,∑_2), then R_1R_2 is simple, if and only if for any finitely many elements t_1, t_2…, t_n ∈∑_2, whenever they are linear ind. over K, they are lin. ind. over ∑_1. This improves a theorem of Azumaya and Nakayama obtained in 1944. Let R_i be K-rings, and L_i be left R_i modules, i=1, 2, in §3, we first define a R_1R_2 mapping φ of L_1×L_2 into a left R_1R_2 module S being multiliner, if Φ(sum from i to u_i a_i, sum from to v_j β_j)=sum from ij(u_iv_j)Φ(a_i, β_j). A left R_1R_2 module S is called a tensor product of L_1 and L_2 written S=L_1L_2, if (1) φ: L_1×L_2 →→ S being multlilinear, and (2) whenever ψ: L_1×L_2 → V being multilinrear, f: S → V, f φ=ψ. The existence, uniqueness, and some fundamental properties of such tensor products are proved. In §4, We considered the functor—L'. Suppose L_μ and L'_R are left R_μ and R'_λ modules. Let σ: R_1 → R'_1 be a K homorphism, f_σ: L_1 → L'_1 be a mapping, such that f_σ(ua+vβ)=uf_σa+vf_σβ∈L'_1, u, v∈R'_1, β∈L_1, we then have the commutative diagram and whenever f_σ.g_τ are isomorbhisms, f_σg_τ is an isomorphism. Some properties of flat modules are considered. If L_μ and L' are projective left R_μ and R' modules, then L_μL' is also projective. If L_μL' is projective, and L' is a projective K module, then L_μ must be projective. Let R be an associative ring, L a left R module, its dual module is a right R module. We then have: if L_μ and L'_λ are finitely generated projective modules, then L_μ~*L'_λ~*≌(L_μL_λ)~*. 20

In the literature, only the tensor product SL of a right R module S and a left R module L is defined and considered, where R is a ring. The theory of such tensor products is very important in the theory of rings, modules, categories and homology, However, such a tensor product is not a ring module, neither a left R module, nor a right R module. In this paper, the tensor product L_1L_2 of two left ring modules is defined, where L_i is a left R_i module, i=1, 2, R_1 and R_2 are K-rings, may not equal. Thus, if...

In the literature, only the tensor product SL of a right R module S and a left R module L is defined and considered, where R is a ring. The theory of such tensor products is very important in the theory of rings, modules, categories and homology, However, such a tensor product is not a ring module, neither a left R module, nor a right R module. In this paper, the tensor product L_1L_2 of two left ring modules is defined, where L_i is a left R_i module, i=1, 2, R_1 and R_2 are K-rings, may not equal. Thus, if is a category of a class of left ring madules L_μ, associated with morphisms H (L_μ, L_λ). L_μ being left R_μ module and if L′is a left R′ module, then —L′ is a functor of. A part of the results obtained are the following: Let K be a commutative ring with unity. In §2, we considered the tensor product of some well-known K rings (wrt. K). Some of them are. If R_1 and R_2 are division rings, K is a field, and ifα∈R_1, β∈R_2, such that they have the same minimum polynomial, over K, then R_1R_2 can′t be a division ring. Let R_1 and R_2 be simble rings over a field K, with centrum ∑_1 and ∑_2 resp. Let F=K(∑_1,∑_2), then R_1R_2 is simple, if and only if for any finitely many elements t_1.t_2…,t_n ∈∑_2, whenever they are linear ind. over K, they are lin. ind. over ∑_1. This improves a theorem of Azumaya and Nakayama obtained in 1944. Let R_i be K-rings, and L_i be left R_1 modules, i=1, 2, in §3, we first define a R_1R_2 mapping φ of L_1×L_2 into a left R_1R_2 module S being multilinear, if A left R_1R_2 module S is calied a tensor product of L_1 and L_2, written S= L_1L_2, if(1) φ:L_1×L_2 →→ S being multilinear, and (2) whenever φ: L_1×L_2 →V being multilinrear, f: S→V, fφ=φ. The exisience, uniqueness, and some fundamental properties of such tensor products are proved. In §4, We considered the functor —L′. Suppose L_μ and L_R are left R_μ and R_λ modules. Let σ:R_1→R_1 be a K homorphism, f_σ:L_1→L_1 be a mapping, such that we then have the commutative diagram and whenever f_σ g)τ are isomorbhisms, f_σg_τ is an isomorphism. Some properties of flat modules are considered. If L_μ and L' are projective left R_μ, and R' modules, then L_μL' is also projective. If L_μL' is projective, and L' is a projective K module, then L_μ must be projective. Let R be an associative ring, L a left R module, its dual module is a right R module. We then have: if L_μ and L'_λ are finitely generated projective modules, then L_μ~*L'_λ~*(L_μL_λ)~*.