- #1

Math Amateur

Gold Member

MHB

- 3,998

- 48

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 someone to check my solution to Problem 2(c) of Problem Set 2.1 ...

Problem 2(c) of Problem Set 2.1 reads as follows:View attachment 8061My attempt at a solution follows:We claim that \(\displaystyle \bigoplus_\Delta R_\alpha\) is a right ideal of \(\displaystyle \prod_\Delta R_\alpha\) Proof ... Let \(\displaystyle (x_\alpha ) , (y_\alpha ) \in \bigoplus_\Delta R_\alpha\) and let \(\displaystyle (r_\alpha ) \in \prod_\Delta R_\alpha\)Then \(\displaystyle (x_\alpha ) + (y_\alpha ) = (x_\alpha + y_\alpha )\) ... by the rule of addition in direct products ...Now ... \(\displaystyle x_\alpha + y_\alpha \in R_\alpha\) for all \(\displaystyle \alpha \in \Delta\) ... by closure of addition in rings ... Thus \(\displaystyle (x_\alpha + y_\alpha ) \in \prod_\Delta R_\alpha\) ...... but also ... since \(\displaystyle (x_\alpha \)) and \(\displaystyle (y_\alpha )\) each have only a finite number of non-zero components ...

... we have that \(\displaystyle (x_\alpha + y_\alpha )\) has only a finite number of non-zero components ...

... so ... \(\displaystyle (x_\alpha + y_\alpha ) \in \bigoplus_\Delta R_\alpha\) ...

Hence \(\displaystyle (x_\alpha ) + (y_\alpha ) \in \bigoplus_\Delta R_\alpha \) ... ... ... ... ... (1)

Now we also have that ... \(\displaystyle (x_\alpha ) (r_\alpha ) = (x_\alpha r_\alpha)\) ... ... rule of multiplication in a direct product ...

Now ... \(\displaystyle x_\alpha r_\alpha \in R_\alpha\) for all \(\displaystyle \alpha \in \Delta\) ... since a ring is closed under multiplication ...

and ...

\(\displaystyle (x_\alpha r_\alpha)\) has only a finite number of non-zero components since \(\displaystyle (x_\alpha )\) has only a finite number of non-zero components ...

So ... \(\displaystyle (x_\alpha r_\alpha) \in \bigoplus_\Delta R_\alpha\)

\(\displaystyle \Longrightarrow (x_\alpha) (r_\alpha) \in \bigoplus_\Delta R_\alpha\) ... ... ... ... ... (2)

\(\displaystyle (1) (2) \Longrightarrow\) \(\displaystyle \bigoplus_\Delta R_\alpha\) is a right ideal of \(\displaystyle \prod_\Delta R_\alpha\)

Can someone please critique my proof ... ... and either confirm its correctness or point out the errors and shortcomings ...

Such help will be much appreciated ...

Peter

I need someone to check my solution to Problem 2(c) of Problem Set 2.1 ...

Problem 2(c) of Problem Set 2.1 reads as follows:View attachment 8061My attempt at a solution follows:We claim that \(\displaystyle \bigoplus_\Delta R_\alpha\) is a right ideal of \(\displaystyle \prod_\Delta R_\alpha\) Proof ... Let \(\displaystyle (x_\alpha ) , (y_\alpha ) \in \bigoplus_\Delta R_\alpha\) and let \(\displaystyle (r_\alpha ) \in \prod_\Delta R_\alpha\)Then \(\displaystyle (x_\alpha ) + (y_\alpha ) = (x_\alpha + y_\alpha )\) ... by the rule of addition in direct products ...Now ... \(\displaystyle x_\alpha + y_\alpha \in R_\alpha\) for all \(\displaystyle \alpha \in \Delta\) ... by closure of addition in rings ... Thus \(\displaystyle (x_\alpha + y_\alpha ) \in \prod_\Delta R_\alpha\) ...... but also ... since \(\displaystyle (x_\alpha \)) and \(\displaystyle (y_\alpha )\) each have only a finite number of non-zero components ...

... we have that \(\displaystyle (x_\alpha + y_\alpha )\) has only a finite number of non-zero components ...

... so ... \(\displaystyle (x_\alpha + y_\alpha ) \in \bigoplus_\Delta R_\alpha\) ...

Hence \(\displaystyle (x_\alpha ) + (y_\alpha ) \in \bigoplus_\Delta R_\alpha \) ... ... ... ... ... (1)

Now we also have that ... \(\displaystyle (x_\alpha ) (r_\alpha ) = (x_\alpha r_\alpha)\) ... ... rule of multiplication in a direct product ...

Now ... \(\displaystyle x_\alpha r_\alpha \in R_\alpha\) for all \(\displaystyle \alpha \in \Delta\) ... since a ring is closed under multiplication ...

and ...

\(\displaystyle (x_\alpha r_\alpha)\) has only a finite number of non-zero components since \(\displaystyle (x_\alpha )\) has only a finite number of non-zero components ...

So ... \(\displaystyle (x_\alpha r_\alpha) \in \bigoplus_\Delta R_\alpha\)

\(\displaystyle \Longrightarrow (x_\alpha) (r_\alpha) \in \bigoplus_\Delta R_\alpha\) ... ... ... ... ... (2)

\(\displaystyle (1) (2) \Longrightarrow\) \(\displaystyle \bigoplus_\Delta R_\alpha\) is a right ideal of \(\displaystyle \prod_\Delta R_\alpha\)

Can someone please critique my proof ... ... and either confirm its correctness or point out the errors and shortcomings ...

Such help will be much appreciated ...

Peter

Last edited: