Writing Assignment 05

Directions.

1. If you want, you may work together with one partner on this and turn in the same pdf for both of you.
   • If you do this, please make sure that you truly collaborate on revising the math and typing it up.
   • To collaborate with Overleaf, one person should start the project, and then use the share option (on the top menu bar), to share it with their partner.
2. Get the template for this assignment (or use your own). Here’s how to do it:
   • Go to www.overleaf.com, and make sure you are logged in.
   • In a new window, go here: https://www.overleaf.com/read/cdwkcphjkcwx#6a356d
   • Click on the menu icon in the upper-left and select “Copy Project”.
   • Create a name for the assignment and click “Copy”.
   • When this completes you will be back in your own workspace (instead of mine).
3. Use $\LaTeX$ to type up your proofs of the Propositions to Prove listed below. Make sure to use complete sentences and appropriate punctuation. Also, make sure to edit for typos.
4. Click on the “Download PDF” button (to the right of the “Recompile” button). Save the file somewhere you can easily find it. Submit the pdf in Canvas.
5. Let me know if you have any questions!

Remember, if you have trouble finding the command for a math symbol you want to use, try googling it, email me, or try looking in this Short Math Guide for $\LaTeX{}$.

Propositions to Prove.

1. Assume $R$ is a commutative ring, and let $a\in R$. Then $\{ar \mid r\in R\}$ is an ideal of $R$. Moreover, if $R$ is a commutative ring with unity, then $\{ar \mid r\in R\}$ is equal to the ideal generated by $a$. (See Theorems 5.82 and 5.91.)

2. Assume $R$ is a commutative ring with unity, and suppose $R$ has at least $2$ elements. If the only ideals of $R$ are $\{0\}$ and $R$, then $R$ is a field. (This is half of Theorem 5.85.)