Modules over an Integral Domain

Newtime

Homework Statement

Let R be a ring with no zero divisors such that for any $$r,s\in R$$, there exist $$a,b \in R$$ such that $$ar+bs=0$$. Prove: $$R=K \oplus L$$ implies $$K=0$$ or $$L=0$$.

Homework Equations

Definition of direct sum of modules, integral domain...

The Attempt at a Solution

I didn't know where to start on this one. In particular, I don't see what the hypothesis has to do with anything and certainly I do not see how it's related to the conclusion. I think all I need is a nudge in the right direction.

Staff Emeritus
Homework Helper
Assume that $$R=K\oplus L$$. Take $$k\in K$$ and $$l\in L$$. Is it ever possible that $$ak=bl$$ for some $$k,l\in R$$??

If this doesn't help you, let's simplify the question: is it possible that k=l?

Newtime
Assume that $$R=K\oplus L$$. Take $$k\in K$$ and $$l\in L$$. Is it ever possible that $$ak=bl$$ for some $$k,l\in R$$??

If this doesn't help you, let's simplify the question: is it possible that k=l?

k=l only if k=l=0 since the sum is direct. Also, it is possible that ak=bl by hypothesis, but I don't see how this points toward a solution.

However, it seems clear that just before we say "thus, l=0" we would use the hypothesis that R is an integral domain and that bl=0.

Staff Emeritus
Homework Helper
Ok, you are correct. k=l only if k=0 and l=0.

Can you describe a similar restriction for ak=bl? I.e. what can you say about ak and bl if ak=bl?

Newtime
Ok, you are correct. k=l only if k=0 and l=0.

Can you describe a similar restriction for ak=bl? I.e. what can you say about ak and bl if ak=bl?

Well the obvious implication is that one is the additive inverse of the other: $$ak-bl=0$$. But this is just a trivial restatement of what you said so I must be missing something here.

Staff Emeritus
Homework Helper
Ok, let's simplify it a bit: can it be true that 2k=l?? (assume for a second that $$2\in R$$). Under what conditions can this be true??

Newtime
Ok, let's simplify it a bit: can it be true that 2k=l?? (assume for a second that $$2\in R$$). Under what conditions can this be true??

Yes but this implies $$k,l \in K \cap L=\lbrace 0 \rbrace$$ does it not?

Staff Emeritus
Homework Helper
Yes, if 2k=l, then k=0 and l=0.

Now, the general case: what if ak=bl??

Newtime
Yes, if 2k=l, then k=0 and l=0.

Now, the general case: what if ak=bl??

I still don't see it. The previous case (2k=l) was obvious because 2k stays in K since of course K is a ring and thus closed under addition (2k=k+k). But a and b are arbitrary elements of R so we don't know where the elements ak and bl end up and I don't think the problem is designed to be broken up into cases. Even so, considering the different subcases doesn't seem to help.

Staff Emeritus
Homework Helper
Maybe you can prove that K and L are in fact ideals if $$R=K\oplus L$$...
So if a is in R, then ak is in K...

Newtime
Maybe you can prove that K and L are in fact ideals if $$R=K\oplus L$$...
So if a is in R, then ak is in K...

I dismissed this idea earlier because I thought it would be more difficult to prove that K and L are ideals than the original claim. But here's what I have :

Let $$K \ni k=(k,0)\in K \oplus L, R \ni r=(k_1,l_1) \in K\oplus L$$. Then, $$rk = (k_1,l_1)(k,0)=(kk_1,0) = kk_1\in K$$. Thus, K is an ideal.

Assuming this is valid, the problem is complete! Thanks for your help.
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://www.physicsforums.com/threads/modules-over-an-integral-domain.469641/", "fetch_time": 1660269235000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2022-33/segments/1659882571538.36/warc/CC-MAIN-20220812014923-20220812044923-00045.warc.gz", "warc_record_offset": 833943115, "warc_record_length": 16164, "token_count": 1005, "char_count": 3390, "score": 3.734375, "int_score": 4, "crawl": "CC-MAIN-2022-33", "snapshot_type": "latest", "language": "en", "language_score": 0.923874020576477, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00033-of-00064.parquet"}