課程大綱

課程資訊

課程名稱

代數專題

Topics in Algebra

開課學期

102-1

授課對象

理學院數學研究所

授課教師

李秋坤

課號

MATH5165

課程識別碼

221 U6110

班次

學分

全/半年

半年

必/選修

選修

上課時間

星期二7,8(14:20~16:20)星期五7(14:20~15:10)

上課地點

天數304

備註

總人數上限：5人

課程簡介影片

核心能力關聯

本課程尚未建立核心能力關連

課程大綱

為確保您我的權利,請尊重智慧財產權及不得非法影印

課程概述

The goal of this course provides the background of studying noncommutative ring theory for master degree students. Our course contains two topics concerning ring theory:
PI-Algebras and Prime GPI-Algebras (a new viewpoint from the theory of functional identities).

(I) PI-Algebras:
1. Kaplansky-Amitsur Theorem
2. Theorem of Amitsur and Levitzki
3. Central simple algebras: converse of Kaplansky-Amitsur Theorem
4. Central polynomials for matrix algebras
5. Structure of Prime PI-algebras
(II) Prime GPI-Algebras
1. Generalized polynomial identities
2. Linear identities
3. Basic functional identities
4. Martindale’s theorem for prime algebras satisfying a generalized polynomial identity
5. Connections between PIs and GPIs: A new viewpoint of Bresar

課程目標

A course of the theory of oncommutative rings (for master degree students )

課程要求

Ring Theory

預期每週課後學習時數

Office Hours

指定閱讀

參考書目

(1) Jacobson, Nathan PI -algebras.An introduction.Lecture Notes in Mathematics, Vol. 441. Springer-Verlag, Berlin-New York, 1975. iv+115 pp.
(2) Jacobson, N. Some recent developments in the theory of PI-algebras. Universale Algebren und Theorie der Radikale, pp. 17–21. Studien zur Algebra und ihre Anwendungen, Band 1, Akademie-Verlag, Berlin, 1976.
(3) Jacobson, N. Some recent developments in the theory of algebras with polynomial identities. Topics in algebra (Proc. 18th Summer Res. Inst., Austral. Math. Soc., Austral. Nat. Univ., Canberra, 1978), pp. 8–46, Lecture Notes in Math., 697, Springer, Berlin, 1978.
(4) Brešar, Matej; Chebotar, Mikhail A.; Martindale, Wallace S., III Functional identities.Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2007. xii+272
(5) Brešar, Matej An alternative approach to the structure theory of PI-rings. Expo. Math. 29 (2011), no. 1, 159–164.
(6) Brešar, Matej A unified approach to the structure theory of PI-rings and GPI-rings, Serdica Math. J. 38 (2012), 199–210

評量方式
(僅供參考)

No.	項目	百分比	說明
1.	期中	50%
2.	期末	50%

課程進度

週次

日期

單元主題