# Submodules of R-Module of Polynomials ....

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In summary: X^n(0 + 0X + 0X^2 + ...)= a_0 + a_1X + ... + a_nX^n + X^2a_2 + X^3a_3 + ... + X^na_n= a_0 + X(a_1 + X(a_2 + ... + X(a_n)))= a_0 + X(a_1 + X(a_2 + ... + X(a_n + 0X^{n+1} + ...)))= a_0 + X(a_1 + X(a_2 + ... + X(a_n + X^{n+1}(0 + 0X + ...))))= a_
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I am reading Paul E. Bland's book: Rings and Their Modules and am currently focused on Section 2.1 Direct Products and Direct Sums ... ...

I need help with Problem 8 of Problem Set 2.1 ...

Problem 8 of Problem Set 2.1 reads as follows:

" 8. Verify Examples 4 and 5 "

I am working on verifying Example 4 ... Example 4 reads as follows: I have tried to verify or prove the assertions in Example 4 as follows:The R-module $$\displaystyle R[X]$$ consists of polynomials of the form

$$\displaystyle p(X) = a_0 + X a_1 + X^2 a_2 + \ ... \ ... \ + X^m a_m$$

where $$\displaystyle X$$ is a commuting indeterminate ... that is $$\displaystyle a X = X a$$ ...Let $$\displaystyle R_k = \{ X^k a_k \mid a_k \in R \}$$
To show $$\displaystyle R_k$$ is a submodule ...

We have that $$\displaystyle R_k \ne \emptyset$$ since $$\displaystyle X^k.1 = X^k \in R_k$$ ... ...

Further ... consider $$\displaystyle p_1 (X) = X^k a_{ k_1 } , \ \ p_2 (X) = X^k a_{ k_2 }$$ ... ...

Now ... we have $$\displaystyle p_1 (X) + p_2 (X) = X^k a_{ k_1 } + X^k a_{ k_2 }$$

$$\displaystyle = X^k ( a_{ k_1 } + a_{ k_2 } )$$

$$\displaystyle = X^k a_{ k_3 }$$ where $$\displaystyle a_{ k_3 } = a_{ k_1 } + a_{ k_2 }$$

$$\displaystyle \in R_k$$ ...Also ... $$\displaystyle p_1 (X) a = ( X^k a_{ k_1 } ) a = X^k ( a_{ k_1 } a ) = X^k a_{ k_4 }$$ where $$\displaystyle a_{ k_4 } \in R$$ ... ... Therefore ... $$\displaystyle R_k$$ is a submodule of $$\displaystyle R[X]$$ ...Is that correct?
Now ... I am unsure of how to write a valid prof of $$\displaystyle R[X] = \bigoplus_{ k = 0 }^{ \infty } R_k$$ ...

... but I record some thoughts ...$$\displaystyle \bigoplus_{ k = 0 }^{ \infty } R_k$$ is an internal direct sum since $$\displaystyle \{ R_k \}_{ k=0 }^{ \infty }$$ is a family of submodules of $$\displaystyle R[X]$$ ...Now $$\displaystyle x \in \bigoplus_{ k = 0 }^{ \infty } R_k = \sum_{ k = 0 }^{ \infty } R_k$$ ...

... is such that $$\displaystyle x = \sum_{ k = 0 }^{ \infty } x_k = \sum_{ k = 0 }^{ \infty } X^k a_k$$ where $$\displaystyle X^k a_k = 0$$ for all but a finite number of $$\displaystyle a_k$$ ...

But this defines all the polynomials in $$\displaystyle R[X]$$ ... so $$\displaystyle R[X] = \bigoplus_{ k = 0 }^{ \infty } R_k$$ ... ...
BUT ... the above is surely a deficient as a "proof" of $$\displaystyle R[X] = \bigoplus_{ k = 0 }^{ \infty } R_k$$ ...Can someone please help me to formulate a valid and convincing proof of $$\displaystyle R[X] = \bigoplus_{ k = 0 }^{ \infty } R_k$$ ...Help will be appreciated ...

Peter

Last edited:
Hello Peter,

Thank you for reaching out for help with verifying Example 4 in Problem Set 2.1 of Paul E. Bland's book. Your approach to proving that R_k is a submodule of R[X] is correct. However, there are a few changes that can be made to your proof of R[X] = \bigoplus_{ k = 0 }^{ \infty } R_k to make it more rigorous and convincing.

First, it would be helpful to define what we mean by R[X] and \bigoplus_{ k = 0 }^{ \infty } R_k. R[X] is the R-module of polynomials over the ring R, and \bigoplus_{ k = 0 }^{ \infty } R_k is the direct sum of the submodules R_k, where each R_k consists of polynomials of degree k or less.

To prove that R[X] = \bigoplus_{ k = 0 }^{ \infty } R_k, we need to show that every element in R[X] can be written as a unique sum of elements from the submodules R_k.

Let x \in R[X] be a polynomial of degree n or less, i.e. x = a_0 + a_1X + ... + a_nX^n. We can write x as a sum of elements from R_k as follows:

x = a_0 + a_1X + ... + a_nX^n = a_0 + a_1X + ... + a_nX^n + 0X^{n+1} + 0X^{n+2} + ...

= a_0 + a_1X + ... + a_nX^n + (0X + 0X^2 + ...) + (0X^2 + 0X^3 + ...) + ...

= a_0 + a_1X + ... + a_nX^n + (0X + 0X^2 + ...) + (0X^2 + 0X^3 + ...) + ... + (0X^n + 0X^{n+1} + ...)

= a_0 + a_1X + ... + a_nX^n + X^2(0 + 0X + 0X^2 + ...) + X^3(0 + 0X + 0X^2 +

## 1. What is an R-module of polynomials?

An R-module of polynomials is a mathematical structure that consists of a set of polynomials with coefficients in a ring R, along with operations for addition and scalar multiplication that satisfy certain properties.

## 2. What are submodules of an R-module of polynomials?

Submodules of an R-module of polynomials are subsets of polynomials that form a smaller R-module within the larger R-module. They still follow the same rules for addition and scalar multiplication as the original module.

## 3. How do you determine if a subset is a submodule of an R-module of polynomials?

In order for a subset to be a submodule of an R-module of polynomials, it must satisfy two conditions: (1) it must be closed under addition, meaning that the sum of any two elements in the subset is also in the subset, and (2) it must be closed under scalar multiplication, meaning that any element in the subset multiplied by a scalar from the ring R is still in the subset.

## 4. What is the significance of submodules in an R-module of polynomials?

Submodules allow us to break down a larger module into smaller, more manageable structures. This can help with understanding and analyzing the properties of the module as a whole.

## 5. Can submodules of an R-module of polynomials have different bases?

Yes, submodules can have different bases from the original module. However, the basis elements of the submodule must also be elements of the basis of the larger module. In other words, the basis of the submodule must be a subset of the basis of the original module.

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