Split Epimorphisms .... Bland Proposition 3.2.4 ....

  • Context: MHB 
  • Thread starter Thread starter Math Amateur
  • Start date Start date
  • Tags Tags
    Split
Click For Summary
SUMMARY

The discussion centers on Proposition 3.2.4 from Paul E. Bland's "Rings and Their Modules," specifically its relationship to Proposition 3.2.3. Proposition 3.2.4 describes a split monomorphism $g':M_2\to M$ with a corresponding splitting map $g$. The clarification provided indicates that understanding the remark following Definition 3.2.2 is crucial for grasping the implications of these propositions. This connection is essential for those studying exact sequences in the context of module theory.

PREREQUISITES
  • Understanding of split monomorphisms in module theory
  • Familiarity with exact sequences in the context of algebra
  • Knowledge of definitions and propositions in "Rings and Their Modules" by Paul E. Bland
  • Basic proficiency in mathematical notation and terminology used in algebra
NEXT STEPS
  • Study the implications of Definition 3.2.2 in "Rings and Their Modules"
  • Review Proposition 3.2.3 and its applications in module theory
  • Explore the concept of exact sequences in greater depth
  • Investigate other examples of split monomorphisms in algebraic structures
USEFUL FOR

Mathematicians, particularly those focusing on algebra and module theory, as well as students seeking to deepen their understanding of exact sequences and their applications in ring theory.

Math Amateur
Gold Member
MHB
Messages
3,920
Reaction score
48
I am reading Paul E. Bland's book "Rings and Their Modules" ...

Currently I am focused on Section 3.2 Exact Sequences in [FONT=MathJax_Main]Mod[FONT=MathJax_Math]R ... ...

I need some help in order to fully understand Proposition 3.2.4 ...

Proposition 3.2.4 reads as follows:
View attachment 8076
Can someone please explain exactly how Proposition 3.2.3 establishes Proposition 3.2.4 ...
Help will be much appreciated ...

Peter
 
Physics news on Phys.org
Hi Peter,

This is a consequence of the remark after definition 3.2.2.

In the case of proposition 3.2.4, $g':M_2\to M$ is a split monomorphism with splitting map $g$, and you can use proposition 3.2.3.
 
castor28 said:
Hi Peter,

This is a consequence of the remark after definition 3.2.2.

In the case of proposition 3.2.4, $g':M_2\to M$ is a split monomorphism with splitting map $g$, and you can use proposition 3.2.3.
Thanks for the help, castor28 ...

Peter
 

Similar threads

Undergrad Split Epimorphisms .... Bland Proposition 3.2.4 ....
  • · Replies 4 ·
Replies
4
Views
2K
MHB Split Monomorphisma .... Bland Definition 3.2.2 & Proposition 3.2.3 ....
  • · Replies 2 ·
Replies
2
Views
1K
Undergrad Split Monomorphisms .... Bland Defn 3.2.2 & Propn 3.2.3 ....
  • · Replies 5 ·
Replies
5
Views
2K
MHB Short Exact Sequences and Direct Sums .... Bland, Proposition 3.2.7 .... ....
  • · Replies 2 ·
Replies
2
Views
2K
MHB Split Short Exact Sequences .... Bland, Proposition 3.2.6 .... ....
  • · Replies 2 ·
Replies
2
Views
1K
MHB Does Proposition 3.2.6 Imply g'(y) = x' - f(f'(x'))?
  • · Replies 1 ·
Replies
1
Views
1K
Undergrad Split Short Exact Sequences .... Bland, Proposition 3.2.6 ....
  • · Replies 3 ·
Replies
3
Views
2K
MHB Direct Sums of Noetherian Modules .... Bland Proposition 4.2.7 .... ....
  • · Replies 2 ·
Replies
2
Views
2K
MHB Principal Ideal Rings and GCDs .... .... Bland Proposition 4.3.3
  • · Replies 1 ·
Replies
1
Views
1K
MHB Understanding Bland's Proof of Proposition 4.3.14: Primitive Elements of Modules
  • · Replies 1 ·
Replies
1
Views
1K