Modules and Spectral Spaces

Prime Submodules and a Sheaf on the Prime Spectra of Modules

“…Proof. Since R is Noetherian, U is quasi-compact by [2,Proposition 3.24]. Thus there exist n ∈ , open subsets W 1 W n of U , t 1 t n ∈ R, l 1 l n ∈ N such that U = n j=1 W j and for each j = 1 n and P ∈ W j we have t j = P M and = l j /t j .…”

“…In 1986, McCasland and Moore adjusted the definition to the M -radical given in [24, p. 37]. Thus, began the extensive study of radical theory for modules, which has continued with the more recent work of many algebraists [1,16,21,23,25,30]. In work beginning in the nineties, many authors have studied analogs of Krulls principal ideal theorem or/and developed a dimension theory for modules [7,12].…”

“…Other analogs for well-known theorems, such as Cohens theorem and the prime avoidance theorem, were given in some papers [15,18]. Also, some new classes of modules such as weak multiplication modules, primeful modules, top modules and strongly top modules are defined by notion of prime submodule (see [1,6,19,27]). The concept of prime submodule has led to the development of topologies on the spectrum of modules.…”

On the Prime Spectrum of Torsion Modules

“…Consequently, the length of any saturated chain of the prime submodules of M starting from P and ending at Q is equal to rank R/p (Q/P ). Afterward, an extensive study of radical theory for modules was begun, which has continued with the more recent work [1,16,20,22,24,28,31]. Many algebraists (for example see [10], [20] and [30]) tried to find a relationship between rad(N ) and √ (N : M ), where N is a submodule of an R-module M .…”

“…In [20], some conditions have been obtained under which rad(N ) = √ (N : M )M , for each submodule N of M . Finding an explicit formula for the radical of submodule is an interesting subject of many papers (see [1,10,19,28]). In the sequel we introduced some expressions for the radical of special submodules of certain modules.…”