This text, drawn from the author's lectures at the University of California at Berkeley, is intended as a textbook for a one-term course in basic ring theory. The material covered includes the Wedderburn-Artin theory of semi-simple rings, Jacobson's theory of the radical representation theory of groups and algebras, prime and semi-prime rings, primitive and semi-primitive rings, division rings, ordered rings, local and semi-local rings, and perfect and semi-perfect rings. By aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation, the author has produced a text which is suitable not only for use in a graduate course, but also for self-study by other interested graduate students. Numerous exercises are also included. This graduate textbook on rings, fields and algebras is intended for graduate students in mathematics.

A first course in noncommutative rings, by T. Y. Lam. Pp. 385. £37 (pb), £62.50 (hb). 2001. ISBN 0 387 95325 6 (pb), 0 387 95183 0 (hb) (Springer-Verlag).

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Infinite Abelian Groups

Determine the value of the variable in each equation.

Basic Algebra = 代数学入門

Commutative rings I

An introduction to noncommutative noetherian rings

In this note all rings are commutative rings with identity and all modules are unital. Let R be a commutative ring with identity. An R-module M is called a multiplication module provided for each submodule N of M there exists an ideal I of R such that N = I M. Various properties of multiplication modules are considered. If there is a common theme it is that the methods used generalise results of Naoum and Hasan proved using matrix methods. 1. Sums and intersections. Let R be a commutative ring with identity and M a unital R-module. The annihilator of M is denoted ann (M) and for any m ~ M the annihilator ofm is denoted ann(m). IfN is a submodule of M then (N: M) denotes the ideal ann(M/N) of R, that is (N :M) = {r e R: rM c= N}. An R-module M is called a multiplication module provided for each submodule N of M there exists an ideal I of R such that N = I M. It is clear that every cyclic R-module is a multiplication module. Let N be a submodule of a multiplication module M. There exists an ideal I of R such that N = I M. Note that I~(N:M) and N=IM=(N:M) M~N so that N=(N:M) M. It follows that an R-module M is a multiplication module if and only if N = (N : M) M for all submodules N of M. An ideal A of R which is a multiplication module is called a multiplication ideal. Let P be a maximal ideal of a ring R. An R-module M is called P-torsion provided for each m ~ M there exists p E P such that (1 - p) m = 0. On the other hand M is called P-cyclic provided there exist x ~ M and q E P such that (1 - q) M ~ R x. Our starting point is the following result taken from [3, Theorem 1.2].

Some remarks on multiplication modules

(1988). Multiplication modules and theorems of mori and mott. Communications in Algebra: Vol. 16, No. 4, pp. 781-796.

Multiplication modules and theorems of mori and mott

(1992). On the prime radical of a module over a commutative ring. Communications in Algebra: Vol. 20, No. 12, pp. 3593-3602.

On the prime radical of a module over a commutative ring

Cyclic modules whose quotients have all complement submodules direct summands

Let R be a ring and M a right R-module. M is called ⌖-supplemented if every submodule of M has a supplement that is a direct summand of M, and M is called completely ⌖-supplemented if every direct summand of M is ⌖-supplemented. In this paper various properties of these modules are developed. It is shown that (1) Any finite direct sum of ⌖-supplemented modules is ⌖-supplemented. (2) If M is ⌖-supplemented and (D3) then M is completely ⌖-supplemented.

Patrick F. Smith

Papers

Some remarks on multiplication modules

Multiplication modules and theorems of mori and mott

On the prime radical of a module over a commutative ring

Cyclic modules whose quotients have all complement submodules direct summands

On ⌖-supplemented Modules