By Goro Azumaya (auth.), Surender Kumar Jain, Sergio R. López-Permouth (eds.)
The papers of this quantity percentage as a standard target the constitution and classi- fication of noncommutative earrings and their modules, and take care of issues of present examine together with: localization, serial earrings, ideal endomorphism jewelry, quantum teams, Morita contexts, generalizations of injectivitiy, and Cartan matrices.
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Extra info for Non-Commutative Ring Theory: Proceedings of a Conference held in Athens, Ohio Sept. 29–30, 1989
Sample text
2), which is circular by minimality of ~2. We m a y select e2 E F and y = ye2 C eiJ("l~) - e~J("/~)J("l~). Since y ~ Q, e~ ~ Q. /~) = e2P2 D %Q and there is a commutative diagram e2Q = e2J(ct) c e2P2 = e2J("/2) eIQ = c = c e2n c v R c_ c eiR. By Prop. 4, P~ ,,-> Pz. This procedure m a y be iterated, producing a sequence of linked p r i m e ideals P1 " > P2 "~> • • • where (by induction and Prop. 5) all P~ are minimal a m o n g prime ideals properly containing Q. Associated with each P~ is the subset Ti = F - P~ _ F .
From L1 < < L2 it follows that D = f i A~-IL~ < < R. , 0 :i/: ~D C_ ML1. Thus ML1 < < ML2. (iii) This follows from (ii) with L = L1 = L2. 47 3. PRIMENESS AND COMPLEMENTS The objective of this section is to study various primeness conditions on submodules K < M and particularly what these conditions imply about the complement closure/t" < M . Not all primeness conditions carry over to complement closures. 13]). 1. P r o p o s i t i o n . 2). Then the following hold. (i) K < M is prime ~ K < M is prime.
Then a ± < ~ R and a ± O C < < R for some 0 # C _< R. From bRaR C_ B A = 0 it follows that also bCaC = 0. Thus CaC C_ b± n C = 0. Since R is semiprime, aCaC = 0, and hence a C = 0. Thus C _c a ± is a contradiction. (ii) ===~ (iii). Let ~ : _~ ---+/3 be an isomorphism, and take any 0 # a 6 A, and hence 0 ¢ ~a 6 B . It follows from Z/~ = 0 and (c}a)-lB < < R that 0 # (~a)[(~a)-lB] = O[a(~a)-lB]. Hence there exists an z • a ( b a ) - l B C_ A with 0 # ~bz = y • B . Since ~b is an isomorphism, z R ~ !