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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

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

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