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I need someone to check my solution to Problem 3(a) of Problem Set 2.1 ...

Problem 3(a) of Problem Set 2.1 reads as follows:https://www.physicsforums.com/attachments/8062

My attempt at a solution follows:We claim that

__right ideal of the ring \(\displaystyle R_1 \times R_2 \times \ ... \ ... \ \times R_n \) is of the form \(\displaystyle A_1 \times A_2 \times \ ... \ ... \ \times A_n\) ...Proof:__

**every**Suppose \(\displaystyle A\) is a right ideal of \(\displaystyle R_1 \times R_2 \times \ ... \ ... \ \times R_n\) ...

Let \(\displaystyle a \in A\) and put \(\displaystyle A_1 = \pi_1 (A) , A_2 = \pi_2 (A) , \ ... \ ... \ , A_n = \pi_n (A)\)

Now \(\displaystyle a \in A\) ...\(\displaystyle \Longrightarrow \pi_1(a) = a_1, \pi_2(a) = a_2, \ ... \ ... \ , \pi_n(a) = a_n\)

for some \(\displaystyle a_1 \in A_1, a_2 \in A_2, \ ... \ ... \ , a_n \in A_n\)Hence ...

\(\displaystyle a = ( i_1 \pi_1 + i_2 \pi_2 + \ ... \ ... \ + i_n \pi_n ) (a) \)\(\displaystyle = (a_1, 0, 0, \ ... \ ... \ , 0) + (0, a_2, 0, \ ... \ ... \ , 0) + \ ... \ ... \ + ( 0, 0, \ ... \ ... \ , a_n ) \)\(\displaystyle = ( a_1, a_2, \ ... \ ... \ , a_n)\)Hence ... \(\displaystyle A \subseteq A_1 \times A_2 \times \ ... \ ... \ \times A_n\) ... ... ... ... ... (1)

*Let \(\displaystyle a_1 \in A_1, a_2 \in A_2, \ ... \ ... \ , a_n \in A_n\)Note that again ... \(\displaystyle A\) is a right ideal of \(\displaystyle R_1 \times R_2 \times \ ... \ ... \ \times R_n\) ...*

**Conversely ...**... and \(\displaystyle a \in A\) and put \(\displaystyle A_1 = \pi_1 (A) , A_2 = \pi_2 (A) , \ ... \ ... \ , A_n = \pi_n (A)\)Then there are \(\displaystyle b_1, b_2, \ ... \ ... \ , b_n \in A\) such that ...

\(\displaystyle \pi_1 (b_1) = a_1, \pi_2 (b_2) = a_2, \ ... \ ... \ , \pi_n (b_n) = a_n \) ... Hence ...

\(\displaystyle b_1 ( 1,0,0, \ ... \ ... \ , 0 ) + b_2 ( 0, 1,0, \ ... \ ... \ , 0 ) + \ ... \ ... \ + b_n ( 0, 0,0, \ ... \ ... \ , 1 ) \)\(\displaystyle = ( i_1 \pi_1 + i_2 \pi_2 + \ ... \ ... \ + i_n \pi_n ) ( b_1 ( 1,0,0, \ ... \ ... \ , 0 ) + b_2 ( 0, 1,0, \ ... \ ... \ , 0 ) + \ ... \ ... \ + b_n ( 0, 0,0, \ ... \ ... \ , 1 ) )\)\(\displaystyle = ( a_1, a_2, \ ... \ ... \ , a_n) \)So ...\(\displaystyle A_1 \times A_2 \times \ ... \ ... \ \times A_n \subseteq A\) ... ... ... ... ... (2)Now ... \(\displaystyle (1), (2) \Longrightarrow A = A_1 \times A_2 \times \ ... \ ... \ \times A_n\)

Can someone please critique my proof ... and either confirm it is correct or point out the errors and shortcomings ... ...

__... there is part of the above proof I do not fully understand ... I will relate the issue to text solution for \(\displaystyle n = 2\) ...In Bland's text on the problem we read the following:__

**Problem/Issue**"... ... Hence \(\displaystyle a(1,0) + b(0,1) = ( i_1 \pi_1 + i_2 \pi_2 ) ( a(1,0) + b(0,1) ) = (a_1, a_2)\), so \(\displaystyle A_1 \times A_2 \subseteq A\). ... ... "I have two questions regarding the above quote:(1) Exactly why/how is the equation \(\displaystyle a(1,0) + b(0,1) = ( i_1 \pi_1 + i_2 \pi_2 ) ( a(1,0) + b(0,1) ) = (a_1, a_2)\) ... true?

Can someone please explain in detail why/how this is true ...

(2) Exactly why/how does the above equation being true imply that \(\displaystyle A_1 \times A_2 \subseteq A\) ... ?

Help with the above will be much appreciated ...

Peter