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

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In summary, the conversation discusses the properties of a countable dimensional vectorspace V over a field F. It is proven that V is a simple R-module, where R denotes End_F V. A basis of V, {e1, e2, ...}, is used to define a module homomorphism φ_j from R to V, which sends f in R to f(e_j). The kernel of φ_j is then found, as well as the Jacobson radical of R. It is then shown that there is exactly one non-trivial twosided ideal in R, defined as ${f ∈ R| dimf (V ) < ∞}$. Finally, it is proven that R is isomorphic to the direct sum of
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peteryellow
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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.
 
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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?
 

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

1. What is an R-module?

An R-module is a mathematical structure in which a ring R acts on an abelian group V, satisfying certain properties such as associativity and distributivity.

2. How do you prove the simplicity of an R-module V?

To prove the simplicity of an R-module V, you need to show that the only submodules of V are the trivial submodule (containing only the zero element) and the entire module V itself. This can be done by showing that any non-trivial submodule must contain the entire module, or by showing that the only ideals of the ring R that are submodules of V are the zero ideal and the entire ring R itself.

3. How do you find the Jacobson radical of a ring R?

The Jacobson radical of a ring R, denoted as J(R), is the intersection of all maximal right ideals of R. It can also be described as the set of all elements in R that annihilate all simple right R-modules. To find J(R), you can first identify all maximal right ideals of R, and then take their intersection.

4. What is an isomorphism between R + R and R?

An isomorphism between R + R and R is a bijective homomorphism (a function that preserves algebraic structure) from the direct sum of two copies of R to R itself. In other words, it is a mapping that preserves addition and multiplication operations between the two structures.

5. How can the simplicity of an R-module V be related to the Jacobson radical of the ring R?

The simplicity of an R-module V is related to the Jacobson radical of the ring R through the concept of simplicity modules. A module V is called simple if it has no proper submodules, and it is called a simplicity module if every submodule of V is either trivial or simple. It can be shown that a module V is a simplicity module if and only if the ring R is a simple ring with J(R) as its radical.

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