Advances in Ring Theory - download pdf or read online

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Hence there exists c = c2 E eR such that E(X) 9:! cR. But this contradicts the hypothesis, HomR(eR, (1 - e)R) = 0. So eR n H = 0, where H is the homogeneous component of X. Since H is an ideal, it follows that (eR)H = O. This contradicts r(eR) = 0. Therefore (1 - e)R contains no nonzero minimal right ideals of R. 5. If R is a right self-injective ring with SOC(RR) right essential in R, then R is a right FI ring. Proof. 2(ii). 5 can be relaxed to "right p-injective and right CS" with virtually the same proof.

Then ¢ = 2:kEA ¢k· Proceeding, now, as in Theorem , for all k E A, we can find Ok, 13k E Q such that ¢k(X) = (Okekk + 13k+1ekk+1)x for all x E I. Observe that, if for some k E A, k - 1 f/. A, then 13k = O. Consequently, for all x E I, ¢(x) = [2:kEA(Okekk + 13k+1ekk+1)] x. Since ¢ -:j; 0, at least one of Ok'S and 13k's is non-zero. Since Q is left maximal ring of quotients of K, there exist 0 -:j; r E K such that for all k E A, ok,13k E K. kYk+1 E J k . ;1 ekk . It is easy to see that RING OF ~IORITA 37 CONTEXT yI::: I.

Let X be a coproduct of X a , a E A. If X is discrete linearly compact, then it is also a product of X a . Proof. For every finite subset I of A put XI = Il(Xa I a E A). The discrete linear compactness of X implies obviously that the canonical morphism X --+ lim{XIXd is an isomorphism. - Remark 3. 1 the product topology on X is equivalent to the discrete topology, although its set of open subobjects does not contain all subobject of X if the direct sum is infinite. In the rest of this example we are going to prove a generalization of Leptin's [6], II.

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