By Jean Michel Lemaire
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Then Pf PN is a module for the Wedderburn ring RfN and hence it is completely reducible. 7 Uj with U3 9:! 7 Qj, where Q3 9:! ::::! ::::! ::::! 7, P 9:! Q and Pis indeed a direct sum of copies of the Pi 's. Finally, suppose P = · L,kEIC Pk with Pk 9:! Pg(k)· Then we have · LPkfPkN=PfPN= · LU3 kEIC . 5. Thus uniqueness is proved D and the result follows, since each Pi = eiR is a cyclic R-module. In view of the preceding, Pi, P 2 , ••• , Pm are, up to isomorphism, the only nonzero indecomposable ·projective R-modules.
Let K be a field and let X be a set of variables. Then the free Kalgebra K(X) is defined just like the polynomial ring K[X] except that the variables do not commute. In particular, K(X) has a K-basis consisting of all formally distinct words in the variables and it follows easily that this ring has no zero divisors. Prove that R = K (X) has IBN by constructing a homomorphism from R to K. Then show that IXI 2: 2 implies that the regular module RR contains a free submodule of infinite rank. Give a proof, directly from the definition, that a weak direct sum of projective modules is projective.
But V/W is clearly generated by W + Vn and is therefore cyclic. Thus (i) now yields the result. 6(i), completes the proof. D We need one more simple observation. 8 Let I be a right ideal of the ring R. i. I = eR for some idempotent e E R if and only if I is a direct summand of RR· Furthermore, when this happens, we have I= el. Part I. Projective Modules 30 ii. If I is a minimal right ideal, then either 1 2 = 0 or I = eR for some idempotent e. PROOF (i) If I= eR, then R = eR+(l-e)R = I+(l-e)R. Conversely, let R =I+ J and write 1 = e + f withe EI and f E J.
Aigebres Connexes et Homologie des Espaces de Lacets by Jean Michel Lemaire