§1. KRS IN AN ADDITIVE CATEGORY 5 Proof. We use induction on m, the case m = 1 being trivial. Assuming m 2, we see that Mm | N1⊕···⊕Nn. By (i) of Lemma 1.2, Mm | Nj for some j by renumbering, we may assume that Mm | Nn. Since Nn is indecomposable and Mm = 0, we must have Mm ∼ Nn. Now (ii) of Lemma 1.2 implies that M1⊕···⊕Mm−1 ∼ N1⊕···⊕Nn−1, and the inductive hypothesis completes the proof. Azumaya actually proved the infinite version of Theorem 1.3: If i∈I Mi ∼ j∈J Nj and the endomorphism ring of each Mi is nc-local, and each Nj is indecomposable, then there is a bijection σ : I −→ J such that Mi ∼ Nσ(i) for each i. (Cf. [Azu48], or see [Fac98, Chapter 2].) We want to find some situations where indecomposables automat- ically have nc-local endomorphism rings. It is well known that idem- potents lift modulo any nil ideal. A typical proof of this fact actually yields the following stronger result, which we will use in the next sec- tion. 1.4. Proposition. Let I be a two-sided ideal of a (possibly non- commutative) ring Λ, and let e be an idempotent of Λ/I. Given any positive integer n, there is an element x ∈ Λ such that x + I = e and x ≡ x2 (mod In). Proof. Start with an arbitrary element u ∈ Λ such that u+I = e, and let v = 1 − u. In the binomial expansion of (u + v)2n−1, let x be the sum of the first n terms: x = u2n−1 + · · · + ( 2n−1 n−1 unvn−1. Putting y = 1 − x (the other half of the expansion), we see that x − x2 = xy ∈ Λ(uv)nΛ. Since uv = u(1 − u) ∈ I, we have x − x2 ∈ In. Here is an easy consequence, which will be needed in Chapter 3: 1.5. Corollary. Let M be an indecomposable object in an additive category A. Assume that idempotents split in A. If E := EndA(M) is left or right Artinian, then E is nc-local. Proof. Since M is indecomposable, E has no non-trivial idempo- tents. Since J (E) is nilpotent, Proposition 1.4 implies that E J (E) has no idempotents either. It follows easily from the Wedderburn-Artin Theorem [Lam91, (3.5)] that E J (E) is a division ring, whence nc- local. 1.6. Corollary. Let R be a commutative Artinian ring. Then KRS holds in the category of finitely generated R-modules. Proof. Let M be an indecomposable finitely generated R-module. By Exercise 1.22 EndR(M) is finitely generated as an R-module and

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