By Hirosi Nagao, Yukio Tsushima
ISBN-10: 0125136609
ISBN-13: 9780125136600
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Rings and Modules summand of V and W => Wu W = W^ © W2 for some W2. We assert that W2 is irreducible. If not, then by the same reason as above, W2 is expressible as a direct sum: W2 = W2 © W"2 (W2, W2 φ 0), and the maximality of W1 implies that (Wl © W2) n(W1® W2)3w. But this is a contradiction since (W1 © W2) n (M^ © 1^2) = ^Ί- Therefore W2 is irreducible as asserted. Now, let {FA; λ e A} be the set of all irreducible A-submodules of V, and let 3~ = {T a A; £ i e r 1^ is a direct sum}. t. inclusion; therefore it contains a maximal element, say T0.
2) => (1). We show that every VA is completely reducible (cf. 1). Let W be any A-submodule of V. Then, since V/W is projective, W is a direct summand of V. 1. ■ Let / be an ideal of a ring A and Ä = A/1. Then, for VA, V = V/VI can be considered as an Ä-module. Also, given an A-homomorphism φ: V -> W, we have an A-homomorphism φ: V -> W(v\->(p(v)). 5. Suppose that P be a projective module. With the notation as above, let λ: P-► V be an Ä-homomorphism. Then there exists an A-homomorphism φ: Ρ -► V such that φ = λ.
A sequence of /4-homomorphism Af-2 Ai-i >Κ·-ι Ai > ^ — ► Ai + i *i+i ►"■ is said to be exact provided I m ^ ! = Ker^· for all i. By a short exact sequence, we mean the exact sequence of the form Ο->ΓΛΚΛΚ"-Ο. This means that λ is a monomorphism and μ is an epimorphism. Thus, V ^ /1(F), Κ/λ(Κ') ~ V" holds. A monomorphism λ: V -► K is said to sp/ii if A(K') is a direct summand of K. Also an epimorphism μ: V-> V" is said to split if K e r ^ is a direct summand of V. 2. Let k:V ->W and μ: W-+V λ o μ = id^.