Prove Simplicity of R-Module V & Find Jac(R) & isomorphism R + R

  • Context: Graduate 
  • Thread starter Thread starter peteryellow
  • Start date Start date
  • Tags Tags
    module
Click For Summary
SUMMARY

The discussion centers on proving that a countably dimensional vector space V over a field F is a simple R-module, where R represents End_F V. The module homomorphism φ_j is defined, mapping elements from R to V via φ_j(f) = f(e_j). The kernel of φ_j is identified, and the Jacobson radical Jac(R) is explored, revealing that the only non-trivial two-sided ideal is the set of functions f in R such that dim(f(V)) < ∞. Furthermore, it is established that R is isomorphic to the direct sum R + R.

PREREQUISITES
  • Understanding of R-modules and vector spaces
  • Familiarity with Endomorphism rings, specifically End_F V
  • Knowledge of module homomorphisms and their properties
  • Concept of Jacobson radical in ring theory
NEXT STEPS
  • Study the properties of simple modules in the context of ring theory
  • Learn about the structure and significance of the Jacobson radical Jac(R)
  • Explore the concept of direct sums in module theory
  • Investigate the implications of isomorphisms in algebraic structures
USEFUL FOR

Mathematicians, algebraists, and graduate students focusing on module theory, ring theory, and linear algebra who seek to deepen their understanding of R-modules and their properties.

peteryellow
Messages
47
Reaction score
0
Let V be a countable dimensional vectorspace over a field F .
Let R denote End_F V .
Prove that V is a simple R-module. If $ e1 , e2 , . . .$ is a basis
of V , then we have a module homomorphism φ_j from R to V ,
sending f in R to f (e_j ).
Find the ker(φ_j) .
Find Jac(R). here I mean jacobson radical.
Prove that there is exactly one non-trivial twosided ideal,
namely
${f ∈ R| dimf (V ) < ∞} $ Prove that R is isomorphic to R + R here + means direct sum . thanks.
 
Physics news on Phys.org


peteryellow, you should know by now that the rules of this forum require you to show us what you've done before we offer any help! So, what have you done?
 

Similar threads

Undergrad Meaning of "identify" & variations in different math context
  • · Replies 5 ·
Replies
5
Views
1K
MHB Proving $\text{GL}_{2}(R): Showing Homomorphism & Isomorphism
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad ##(A/\mathfrak{a})_{\mathfrak{p}/\mathfrak{a}}## and its isomorphism?
  • · Replies 31 ·
2
Replies
31
Views
3K
Graduate Field on R^3 and isomorphism between C and R
  • · Replies 5 ·
Replies
5
Views
4K
Graduate Is the Rank-Nullity Theorem Always True for Linear Operators?
  • · Replies 3 ·
Replies
3
Views
5K
MHB Proving Field Isomorphism: $\sigma (x) = x$ for $x \in K$
  • · Replies 9 ·
Replies
9
Views
3K
Undergrad Is $\Bbb Z\otimes_{\Bbb Z} \Bbb R$ isomorphic to $\Bbb C$?
  • · Replies 2 ·
Replies
2
Views
3K
Graduate How Is Symmetric Algebra Isomorphic to a Free Commutative R-Algebra?
  • · Replies 2 ·
Replies
2
Views
3K
Proving a mapping from Hom(V,V) to Hom(V*,V*) is isomorphic
  • · Replies 3 ·
Replies
3
Views
2K
How to show G/Z(R(G)) is isomorphic to Aut(R(G))?
  • · Replies 7 ·
Replies
7
Views
2K