环与模的素根和素子模.pdfVIP

The prime radical and submodules on Rings and Modules RM M R.LMcCasla M.E.Moore rad(I ) V (I ) R R Mc.Casland I Abstract Since the conception of the prime submodule was introduced, it was researched widely. Being similar to research the prime ideals, we can investigate the prime submodules from the prime radical of the modules, which is the intersection of all prime submodules of the module. We have known that the prime radical of rings has a certain form, but it’s difficult for the prime radical of modules. Mc.Casland and Moore show [14] that there is a certain form of the prime radical of a finitely generated module on principal ideal domains (PID). In this thesis, we investigate the certain form of the prime radical of special modules on rings or modules on special rings. In the fist chapter, we mainly introduce some background and significance of our research. In the second chapter, we mainly research the properties of operation of radical ideals. Some results in commutative algebra have been given from the intersection of prime ideals. In this chapter we research the prime radical ideal from the multiplication of the ideals. Finally, we research the relation between rad(I ) and V (I) . In the third chapter, we research the prime radical of rings and point out that the certain form of prime radical of some rings. Next we discuss the certain form of prime radical of modules in two ways. One is to research the modules on any rings, which have some certain condition such that its prime radical has the certain form. The other mainly research the modules on domains. In the fourth chapter, we investigate the prime submodules on free modules. Mc.Casland had referred to them ([15]). In this chapter we main research them from the natural homomorphism and prime chain of submodules. Key words: Prime ideal Prime radical ideal Variety of indeals Prime submodule Prime chain II Abstract III R R E.W.Swokowask R Feller NR M A.W.Goldie MR E.H.Feller RM M M (N : M ) = {r ∈ R rM ?