Quantale中的素根定理

Prime Radical Theorem in Quantales

在Quantale中引入了m系的概念,利用m系讨论了Quantale中素理想和半素理想之间的关系。在此基础上证明了当Q是可换Quantale时,Id(Q)是空间式Quantale当且仅当Q中的任一理想都是半素理想。最后把环和序半群中的素根定理推广到Quantale中,得到了Quantale中的素根定理。

In this paper,the concept of m system in a quantale Q is introduced.We use the m system to discuss the relations between prime ideals and semiprime ideals in the quantale Q.Based on these,we prove that Id（Q） is a spatial quantale if and only if every ideal in Q is semiprime,if Q is a commutive quantale.Finally,the prime radical theorem in rings and ordered semigroups is generalized in quantales.Hence we obtain the prime radical theorem in quantales.