1. M. Noether, Math. Ann.6, (1873), p. 351.
2. a) ?The Theorem of Residuation, etc.? (Proc. London Math. Soc. (1) 31 (1900), p. 381). The following have reference to this paper: b) C. A. Scott, ?On a Method for dealing with the Intersections of Plane Curves? (Trans. Am. Math. Soc.3 (1902), p. 216). c) F. S. Macaulay, Same title and publication as the preceding (5 (1904), p. 385). d) F. S. Macaulay, ?The Intersections of Plane Curves, with Extensions ton-dimensional Algebraic Manifolds' (Verhandlungen des III. Internationalen Math. Kongresses, B. G. Teubner (1905), p. 284). The proof in VII, p. 307, is wrong, and the properties there stated to be true for any space-curve are only true for a space-curve of the principal class (§ 56 below). e) E. Löffler, ?Zum Noetherschen Fundamentalsatz? (Math. Ann. 65 (1908), p. 400). The following sourees are specially used in the present paper: f)(I) D. Hilbert, ?Über die Theorie der algebraischen Formen? (Math. Ann.36, (1890), p. 473) and f)(II) D. Hilbert, ?Ein allgemeines Theorem über algebraische Formen? (Math. Ann. 42 (1893), p. 320 [§ 3 of the Memoir: ?Über die vollen Invariantensysteme?]), an extension of a theorem due to Netto (Acta Math. 7 (1886), p. 101). g) J. König, ?Einleitung in die allgemeine Theorie der algebraischen Größen? (B. G. Teubner, 1903), for Kronecker's, Theory of the Resolvent and the general theory of Modular Systems. (For continuation see next page.) h) E. Lasker, ?Zur Theorie der Moduln und Ideale? (Math. Ann. 60 (1905), Satz VII, p. 51; Satz XXVII, p. 95; and § 45, p. 98?103). Previous nomenclature has been departed from in the following instances: ?prime equation ? has been replaced by ?principal equation? (to aviod any suggested special connection with prime mudule); ?one-set system? has been replaced by ?system with a single principal equation?; and ?linear equation? has been replaced by ?modular equation? (§16), which is not to be understood as meaning a congruence equation.
3. E. H. Moore (Bull. Am. Math. Soc. (2) 3 (1897), p. 372) defines simple module in this sense, and we shall not use it in any other sense.
4. The process is one which can be actually carried out for any K-N-module whose basis is given, and the value of ? found. It is not possible to give a formula for ? except in a few specially simple cases the most important being the following obvious extension ton variables of Noether's result, for two curves (Math. Ann.6 (1873), p. 351): If the resultant of the terms of lowest degreei 1,i 2,...,i n inF 1,F 2, ...,F n respectively is not zero, then the characteristic number ? of the simple N-module contained in (F 1,F 2, ...,F n) isi 1+i 2+...+i n?n+1. See also Bertini (Math. Ann. 34 (1889), p. 447) who gives a superior limit for ? in the case of two curves when the resultant spoken of is zero. This result can also be obviously extended ton variables.