18.705 (fall 2012): SYLLABUS (running)

Please see the course home page for important additional information about the course.

The following assignments refer to lecture notes written in collaboration with Allen Altman. These notes are available here in three formats: the first implements hyperreferencing in xdvi; the second, in acroread; and the third saves trees when printed.

NOTES (current 12/09/03): 12Nts.dvi and 12Nts.pdf and 12Nts-2up.pdf.

The notes are also available at the URL

http://www.centerofmathematics.com/wwcomstore/index.php/commalg.html

in digital and print forms. The digital form is free of charge; the print form is a perfect-bound paperback, and costs $19.95.

Assigned problems are indicated below, after the date they're due. The numbers between parentheses refer to subsections, which are exercises, in the lecture notes. Note that all the exercises are solved in the notes.

The notes also include solutions to the unassigned problems. Do try to solve each of them too, and do so before reading its solution, in order to better appreciate the issue. And do read the solution even if you think you already know it, just to make sure. Further, some problems have alternative solutions, which may enlighten you.

The problems are exercises for your mathematical health. They provide a means for you to check, solidify, and expand your understanding of the material. They are not meant to be difficult, or tricky, or involved. Rarely do they introduce any new methods of proof.

If you find that you are stuck on any problem, then review the relevant material. If you remain stuck, then discuss the problem with someone or look up its solution. You are on your honor to think through each problem on your own, and to write it up in your own words, unaided by other people, solutions, or notes. Remember: in the end, what counts is that you can solve the problem correctly and that you can explain the solution clearly.

The assigned problems will form the basis for your grade. Your written solutions will count 40%. The remaining 60% will reflect your performance on the midterm (20%) and final (40%), where you will be asked to solve to a selection of the assigned problems.

1. R.9/6.--- Rings and Ideals. Write up for 9/13: (1.6); (1.9); (1.13); (1.15); (1.16).
2. T.9/11.--- Prime Ideals. Write up for 9/20: (2.2); (2.10); (2.18); (2.20); (2.21); (2.27).
3. R.9/13.--- Radicals. Write up for 9/20: (3.6); (3.10); (3.11); (3.13); (3.17); (3.20).
4. T.9/18.--- Modules. Write up for 9/27: (4.3); (4.12); (4.14); (4.15); (4.16).
5. R.9/20.--- Exact Sequences. Write up for 9/27: (5.5); (5.6); (5.16); (5.21); (5.27).
6. T.9/25.--- Direct Limits. Write up for 10/4: (6.5)(1)-(3); (6.5)(4)-(5); (6.9); (6.16)(1); (6.16)(2).
7. R.9/27.--- Filtered Direct Limits. Write up for 10/4: (7.2); (7.6); (7.9); (7.11); (7.14).
8. T.10/2.--- Tensor Products. Write up for 10/11: (8.6); (8.8); (8.13); (8.14); (8.16).
9. R.10/4.--- Flatness. Write up for 10/11: (9.7); (9.9); (9.10); (9.15); (9.19); (9.22).
10. R.10/11.--- Cayley--Hamilton Theorem. Write up for 10/18: (10.6); (10.9); (10.13); (10.14); (10.17); (10.30).
11. T.10/16.--- Localization of Rings. Write up for 10/25: (11.3); (11.8); (11.9); (11.16); (11.24); (11.25).
12. R.10/18.--- Midterm covering through the Cayley--Hamilton Theorem.
13. T.10/23.--- Localization of Modules. Write up for 11/1: (12.4); (12.5); (12.6); (12.7); (12.11); (12.13).
14. R.10/25.--- Support. Write up for 11/1: (13.2); (13.5); (13.15); (13.17); (13.18); (13.21); (13.28).
15. T.10/30.--- Krull--Cohen--Seidenberg Theory. Write up for 11/08: (14.4); (14.5); (14.6); (14.12); (14.14); (14.17).
16. R.11/01.--- Noether Normalization. Write up for 11/08: (15.7); (15.11); (15.16); (15.17); (15.25); (15.26).
17. T.11/06.--- Chain Conditions. Write up for 11/15: (16.2); (16.8); (16.17); (16.20); (16.24); (16.26).
18. R.11/08.--- Associated Primes. Write up for 11/15: (17.6); (17.7); (17.10); (17.15); (17.22); (17.23).
19. T.11/13.--- Primary Decomposition. Write up for 11/29: (18.7); (18.8); (18.17); (18.21); (18.25); (18.26).
20. R.11/15.--- Length. Write up for 11/29: (19.2); (19.4); (19.5); (19.8); (19.10); (19.18).
21. T.11/20.--- Hilbert Functions. Write up for 11/29: (20.5); (20.6); (20.9); (20.10); (20.15); (20.19).
22. T.11/27.--- Dimension. Write up for 12/06: (21.8); (21.12); (21.13); (21.14); (21.16); (21.21).
23. R.11/29.--- Completion. Write up for 12/06: (22.9); (22.10); (22.12); (22.21); (22.24); (22.28).
24. T.12/04.--- Discrete Valuation Rings. Optional, but for extra credit, write up for 12/07: (23.6); (23.7); (23.8); (23.10); (23.15); (23.20).
25. R.12/06.--- Dedekind Domains. Optional, but for extra credit, write up for 12/07: (24.5); (24.6); (24.8); (24.12).
26. T.12/11.--- Fractional Ideals.