10 1. THE KRULL-REMAK-SCHMIDT THEOREM 1.18. Exercise. Let M be an object in an additive category. Show that every direct-sum (i.e., coproduct) decomposition M = M1 ⊕ M2 has a biproduct structure. 1.19. Exercise. Let M be an object in an additive category. (i) Suppose that M has either the ascending chain condition or the descending chain condition on direct summands. Prove that M has an indecomposable direct summand. (ii) Prove that M is a direct sum (biproduct) of finitely many in- decomposable objects. 1.20. Exercise. Prove Steinitz’s Theorem ([Ste11]): Let I and J be non-zero fractional ideals of a Dedekind domain D. Then I ⊕ J ∼ D ⊕ IJ. 1.21. Exercise. Let Λ be a ring with 1 = 0. Prove that the fol- lowing conditions are equivalent: (i) Λ is nc-local. (ii) J (Λ) is the set of non-units of Λ. (iii) The set of non-units of Λ is closed under addition. (Warning: In a non-commutative ring one can have non-units x and y such that xy = 1.) 1.22. Exercise. Let M and N be finitely generated modules over a commutative Noetherian ring R. Prove that HomR(M, N) is finitely generated as an R-module. 1.23. Exercise. Let (R, m) be a Henselian local ring and X, Y , M finitely generated R-modules. Let α: X −→ M and β : Y −→ M be homomorphisms which are not split surjections. Prove that [α β]: X ⊕ Y −→ M is not a split surjection. 1.24. Exercise. Let M be a finitely generated module over a com- plete local ring (R, m). Show that M is complete for the topology defined by the submodules mnM, n 1. 1.25. Exercise. Prove Fitting’s Lemma: Let Λ be any ring and M a Λ-module of finite length n. If f ∈ EndΛ(M), then M = ker(f n ) ⊕ f n (M). Conclude that if M is indecomposable then every non-unit of EndΛ(M) is nilpotent. 1.26. Exercise. Use Exercise 1.21 and Fitting’s Lemma from the exercise above to prove that the endomorphism ring of any indecom- posable finite-length module is nc-local. Thus, over any ring R, KRS holds for the category of left R-modules of finite length. (Be careful:

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