By Donald S. Passman

ISBN-10: 0821836803

ISBN-13: 9780821836804

First released in 1991, this ebook includes the center fabric for an undergraduate first direction in ring idea. utilizing the underlying topic of projective and injective modules, the writer touches upon quite a few features of commutative and noncommutative ring concept. specifically, a few significant effects are highlighted and proved. half I, 'Projective Modules', starts off with uncomplicated module concept after which proceeds to surveying quite a few distinct sessions of earrings (Wedderbum, Artinian and Noetherian jewelry, hereditary jewelry, Dedekind domain names, etc.). This half concludes with an creation and dialogue of the recommendations of the projective dimension.Part II, 'Polynomial Rings', stories those jewelry in a mildly noncommutative environment. many of the effects proved contain the Hilbert Syzygy Theorem (in the commutative case) and the Hilbert Nullstellensatz (for virtually commutative rings). half III, 'Injective Modules', comprises, particularly, a variety of notions of the hoop of quotients, the Goldie Theorems, and the characterization of the injective modules over Noetherian earrings. The publication includes a number of workouts and a listing of urged extra examining. it really is compatible for graduate scholars and researchers drawn to ring idea.

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2(i), W is naturally a simple R-module. Conversely, if V is an irreducible R-module, then by the preceding two results, V ':f: V N. 2(ii), Vis naturally a simple R/N-module. 2(i)(ii), it follows that there is a one-to-one correspondence between the isomorphism classes of irredueible R-modules and of irreducible R/N-modules. 5(i), with representatives Vi, V2, ... , Vm, for instance. Now let Ube a completely reducible R-module. ::! 1 I f(j) i} depends only on U and Vi. To this end, = = Chapter 5.

Later in this chapter we discuss the more important Jacobson radical. An ideal I of R is said to be nil if every element of I is nilpotent. In particular, any nilpotent ideal is nil. Furthermore, as we have observed earlier, if J is a nilpotent right ideal of R, then RJ is a nilpotent, and therefore a nil, two-sided ideal. 1 Let I be a nil ideal of R. i. Ihc E R is nilpotent, for example if x E I, then 1- x is invertible. ii. If J /I is a nil ideal of R/ I, then J is a nil ideal of R. iii. An arbitrary sum of nil ideals is nil.

It remains to show that N is nilpotent, since N = Nil(R) certainly contains all nilpotent ideals of R. Since R is Artinian, the descending chain N 2 N 2 2 N 3 2 · · · must stabilize. Thus suppose Nk = Nk+l and let I = { r E R I r Nk = 0 }. Then I is easily seen to be a right ideal of R and the goal is to show that 46 Part I. Projective Modules I = R. If this is not the case, then the minimum condition applied to the set of right ideals of R properly larger than I yields a minimal such right ideal J.

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