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By Fokkinga M.M.

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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.