By John W. Bunce, F. S. Van Vleck, James W. Brewer

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**Example text**

Tion is: Another equivalent formula An R-module M is noetherian if and only if it satisfies the maximum con dition — that is, each nonempty family of submodules of M contains a maximal element. Before proving our main result on noetherian modules, we require two lemmas. 3. Let R be a commutative ring with M an R-module. If A and B are submodules of M such that A/B and (A + B)/A are finitely generated, then B is finitely generated. Proof. For m E M, let Ш denote the coset m + A. ,b„ generate (A + B)/A and Ь„+ 1 >•••>Ь^ generate A n в.

The system is said to be a s in g le in pu t system when m = I and a sin g le -o u tp u t system when p = I.

20. 20 often in subsequent chapters. 20, but to prove it, we require the following fact. Recall that a module A over an integral domain D is called to r s io n - f r e e if, for each d G D, a G A, from da = 0 it follows that d = 0 or a = 0. The result we need, but only state, is: A finitely generated torsion-free module over an integral domain I. A lgebraic P relim in aries 35 D is isomorphic to a submodule of a free D-module. 21. Let D be an integral domain. Then D is a Bézout domain if and only if each finitely generated torsion-free D-module is free.