非结合非分配环的Jacobson根及在极小条件下半单纯非结合非分配环的结构

本文对非结合非分配环(以下简称两非环)引进Jacobson根概念,同时证明了它是文中意义下的极大合格正则右理想之交,并且通过一系列概念及结果,主要来建立两非环的结构定理,任何满足右理想极小条件的半单纯两非环R只有有限多个单纯理想,并且R是这些单纯理想之直和,这些单纯理想都是满足右理想极小条件的单纯半单纯两非环,它们中的每一个都可分解成有限多个极小右理想之直和,特别两非环取为通常结合环时,本文的结果包含通常结合环所熟知的结果。

JACOBSON-RADiCAL OF. BOTH NON-ASSOCIATIVE AND NON-DISTRIBUTIVE RINGS AND THE STRUCTURE OF SEMI-SIMPLE NON-ASSOCIATIVEAND NON-DISTRIBUTIVE RINGS SATISFYING THE MINIMAL CONDITION FOR RIGHT IDEALS

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.