- #1
Math Amateur
Gold Member
MHB
- 3,998
- 48
I am reading Paul E. Bland's book, "Rings and Their Modules".
I am trying to understand Chapter 4, Section 4.2 on Noetherian and Artinian Modules and need help with fully understanding Proposition 4.2.3, particularly assertion (2).
Bland's statement of Proposition 4.2.3 reads as follows:
View attachment 3733
Consider now Figure 1 below, showing module M with three submodules, and respectively:
View attachment 3734
As per Bland's assertion (2) of Proposition 4.2.3 above, the collection of submodules when ordered by inclusion, clearly has a maximal element, namely .BUT ... ... what is the situation when we consider the collection ?Are both and respectively, each maximal elements in the collection ?
(as you may see from the above I am somewhat unsure as to how to view collections which include disjoint submodules!)
Can someone please help clarify the above issue?
Further to the above analysis, would the module M shown in Figure 1 be noetherian?
It seems to me that M would be noetherian since the only chains of submodules, namely
and
clearly terminate ... ... Can someone please indicate whether this analysis is correct?
Peter
I am trying to understand Chapter 4, Section 4.2 on Noetherian and Artinian Modules and need help with fully understanding Proposition 4.2.3, particularly assertion (2).
Bland's statement of Proposition 4.2.3 reads as follows:
View attachment 3733
Consider now Figure 1 below, showing module M with three submodules,
View attachment 3734
As per Bland's assertion (2) of Proposition 4.2.3 above, the collection of submodules
(as you may see from the above I am somewhat unsure as to how to view collections which include disjoint submodules!)
Can someone please help clarify the above issue?
Further to the above analysis, would the module M shown in Figure 1 be noetherian?
It seems to me that M would be noetherian since the only chains of submodules, namely
and
clearly terminate ... ... Can someone please indicate whether this analysis is correct?
Peter