Submodules of R-Module of Polynomials ....

[Math Amateur]

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

 

[jvicens]

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 +