¹⁾ We take this opportunity to correct the following errata: in the sixth line following the proof of Theorem 16 (page 537) both should be replaced by .

²⁾ S. K. Lemma 6.

³⁾ Cf. Corollary 1 to Theorem 15; there the assumption that the unit element is the only non-zero idempotent element is superfluous; it follows automatically.

⁴⁾ = condition B) in S. K. Theorem 19.

⁵⁾ = non-automorphisms = non-regular elements in the endomorphism ring of m_μ.

⁶⁾ This theorem can readily be transferred, together with Lemma 1, to non-commutative groups if we consider “normal endomorphisms” and make use of the notion of their “Addierbarkeit.” Cf, footnote (27) in S. K.

⁷⁾ The validity of this lemma and that of the succeeding theorem under the assumption (*) alone were communicated to the writer by T. Nakayama.

⁸⁾ Generally, if m is a directly indecomposable module satisfying the same assumption as (*) and if {a ₀} be a summable system of proper endomorphisms of m then the sum is also proper.

⁹⁾ This theorem was given in S. K. Theorem 20, i) (in its proof, to be precise) and iii) by assuming not only the assumption (*) but also the assumption (**) below.

¹⁰⁾ Cf. Corollary to S. K. Theorem 11.

¹¹⁾ Take account of footnote 8).

¹²⁾ Because for an arbitrary non-zero element u of m _μ ue _μ ae _v vanishes for almost all v.

¹³⁾ Cf. S. K. Theorem 8.

¹⁴⁾ Cf. S. K. Lemma 8.

¹⁵⁾ = condition A) in S. K. Theorem 19.

¹⁶⁾ But in case M is finite a is always an automorphism, as Theorem 2, ii) shows.

¹⁷⁾ We may take R ₀ to be a valution ring of a p-adic number field, for instance.

¹⁸⁾ Cf. S. K. Theorem 22.