【摘要】 <正> §1.本文的目的是在對於所謂素性環（Primal Ring）作一些探討.這裹的環都是指着有么元無零因子的可換環.我們以R表這樣一個環,1就是R的么元,大寫字母A,B,C,P,……表R的真理想子環,小寫字母a,b,c,x,y等表R的元.符號Ax^-1表示R中一切能使xy∈A的y所組成的集.容易證明Ax^-1是一個理想子環,並且Ax^-1A.如果Ax^-1A,則說x不素於A,否則說x素於A.這樣一來,A是素理想子環的充要條件就是R中凡不屬A的元都素於A.更多还原

【Abstract】 Only commutative rings with unity and no divisor of zero are considered. Such a ring is called a primal ring if every ideal of it is primal in the sense of L. Fuchs. The author showed that if R is a primal ring, then （1） The prime ideals of R are simply ordered; （2） R has at most one prime principal ideal （in addition to R itself）; （3） the set of all non-units of R is a prime ideal P of R, and in case R has a prime principal ideal, this ideal is P; （4） the ideal P is the adjoint prime ideal of every principal ideal. The following theorems have also been proved.Theorem 1. R is a valuation ring, if and only if R is primal and every non-principal ideal cannot be generated by a finite number of elements.Theorem 2. R is a subring of a valuation ring Q of the quotient field K of R such that the set of all non-units of R coincides with the maximum prime ideal of Q if and only if R is primal such that whenever a, b ∈ R, a + b and b + a, the quotients （a） b^-1 and （b） a^-1 always equal to the maximum prime ideal of R.Theorem 3. An integral domain R is primal, if and only if the set of all prime ideals of R is simply ordered by inclusion.Theorem 4. A unique factorization domain R is primal, if and only if it is a valuation ring of its quotient field with respect to some discrete archimedean valuation.更多还原