I am reading a graduate-level Abstract Algebra lemma on noetherian integral domain, I am bring it up here hoping for pointers. The original passage is in one big-fat paragraph but I broke it down here for your easy reading. Let me know if I forget to include any underlying lemmas, thank you for your time and help. (I posted this in homework forum but apparently it was not the right place since I did not get any response at all.)(adsbygoogle = window.adsbygoogle || []).push({});

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

LEMMA:

Let M be anR-module. LetTbe maximal among the ideals ofRsuch thatMpossesses a submoduleLfor whichL/LTis not noetherian. ThenTis a prime ideal ofR.

PROOF:

(1) We are assuming thatMpossesses a submoduleLfor whichL/LTis not noetherian. Thus, asL/LR = L/Lis noetherian,T≠R.

(2) Let us assume, by way of contradiction, thatTis not prime. ThenRpossesses idealsUandVsuch thatT ⊂ U,T ⊂ V, andUV ⊆ T.

(3) The (maximal) choice of T forcesL/LUandLU/LUVto be noetherian. [QUESTION:I understand that whileTis maximal butUandVare strictly larger thanT, and the only way to resolve this paradox is to takeUandVas structures different fromT. But I am lost on how all these "forceL/LUandLU/LUVto be noetherian."]

(4) Thus, by Lemma below,L/LUVis noetherian. [QUESTION: Does it mean that sinceL/LUandLU/LUVare noetherian from above, thereforeL, LUandLUVare noetherian, and thereforeL/LUVis noetherian?]

(5) On the other hand, asUV ⊆ T, LUV ⊆ LT. Thus,L/LTis a factor module ofL/LUV. [QUESTION: Here, I am begging explanation on howL/LTis a factor module ofL/LUV,step-by-step if possible.]

(6) Thus, asL/LUVis noetherian,L/LTis noetherian; cf. Lemma below. This contradiction finishes the proof.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

This is the lemma quoted above: LetMbe anR-module, and letLbe a submodule ofM. ThenMis noetherian if and only ifLandM/Lare noetherian.

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Prime Ideal & Noetherian Integral Domain

Tags:

Loading...

Similar Threads - Prime Ideal Noetherian | Date |
---|---|

I Prime and Maximal Ideals in PIDs .. Rotman, AMA Theorem 5.12 | Sep 1, 2016 |

Prime ideal (x) in k[x,y] | Nov 12, 2013 |

Primary Ideals, prime ideals and maximal ideals - D&F Section 15.2 | Nov 9, 2013 |

Primary Ideals, prime ideals and maximal ideals. | Nov 8, 2013 |

Prime Ideals | Jan 7, 2013 |

**Physics Forums - The Fusion of Science and Community**