By V. Kharchenko

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**Extra resources for Automorphisms and derivations of associative rings**

**Example text**

Jl ~ E R, ~ ' ) 2 . , E (II rJl E Q and Jl is a derivation of the ring Q. And, finally, the uniqueness of the extension follows from the equalities where j E (II r)2. The lemma is proved. 2. It follows from the lemma proved above that any derivation of the ring R has a unique extension on Q and R,.. From now on we shall therefore consider that the derivations are determined on Q. Let us introduce the following notation: D (R ) = (Jl E Der Q I 3 I E F, I Jl s:: R). (15) It is clear that the derivations of the ring R are contained in D ( R).

Corollary. ", r)\ rE R}. Proof. It suffices to assume that in the preceding theorem the group acts trivially. 9. Let us consider a 19 CHAPTER 1 I ; II, formal n x n matrix e= with all its points occupied by ; , where n is the order of the group G. Since nR = R and nx = 0 ~ x = 0 in the ring R, then the multiplication of matrices from Rn by e is defined, in which case e2 = e both as a matrix and an operator. • , eame over . • , r gn.. II rE R): Multiplying this equality from the right and from the left bye, we see that the ring eRne is quite integer over the subring eTe of a certain degree m.

The subrings Q, C are closed in RF and. hence. they are complete modules over c. Proof. Let r= lim ra and earaIa(;;. R, eaIa s;;: R, where Ia E F. Then J = LeaIa E F, in which case reaIa = raeaIa' and, hence, rJ (;;. , r E Q. The subring C is closed as an intersection of all the kernels of quite continuous mappings ad a: x ~ xa - ax. The lemma is proved. 15. Let us recall that the module M is called projective if it has the following property. Let 1r be the epimorphism of a module B on a module A, then any homomorphism qJ: M ~ A can be "raised" to the homomorphism",: M ~ B, such that 1r.