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Additional info for Automorphisms and derivations of associative rings
Indeed, if I 2 r = 0, then 0 = II(II aA)g, where A is such a 4} CHAPTER } set that A g= I} r. This set exists since I} is an ideal and I} ~ I g . , r = 0, which fact proves that I2 Let us now define the mapping a g: I2 ~ i E I 2 , then i =j g for a certain j E :? I a I, R E F . in the following way. If and we set ia g = ( ja) g , in which the right part of the equality is determined because ja ~ I. 9). We can now easily check if a ~ a g is the sought extension of g. In this case the formula other extension, j gag = (ja ) g g I, we j gag - j g a g'= ( ja)g - (ja)g = 0, g = g.
Any ideal I of a semiprime ring R has a zero intersection with its annihilator ann I. The direct sum I+ annI belongs to IF. Proof. The intersection III annI has a zero multiplication and, hence, equals zero. If (I + ann I)x =0, then Ix = 0 and, hence, x belongs to the ideal A= ann I. , xE All annA= 0, which is the required proof. 4. Definition. An ideal of the ring R is called essential if it has a zero intersection with any nonzero ideal of the ring R. 5. Lemma. The ideal I of a semiprime ring R belongs to IF iff it is essential.
By induction over quite integer over T of a certain degree m. Let k = 1 and a 1= diag( rl'0, •. O)E ~. ')' Then for k n - k rows and n - k we will show R to be Let us find an element any ~E ~ we have (a1 - t) ~ = 0, and, hence, at a 2 = ta 2 which is the required proof. Let k> 1. Let us present an arbitrary matrix aE Rk as a= [ 0] * * ° °°° a' r" (7) where a' is the (k - 1) x (k - 1) matrix, k - 1. Let us set (a'O) iI= 00 at = ~ E R k _ t' Then for all the elements at, r'i iff the columns and r'i ~E Rk we shall introduce the relation corresponding to the matrices at and r;.