Let 7? be a commutative ring, and E an 7?-module. For each £^1, we define the ideal ap(E) to be the annihilator of the pth exterior product of E (denoted by A" E), and ap(E) is called the pth invariant factor of E. If £ is a finitely generated 7?-module, then Ap £ = 0 (or «"(£) =7?) for all but a finite number of p, and the largest integer p for which l\p £^0 is called the exterior rank of £ (denoted by ext rank £). The fact that the invariant factors of a module give some insight into the

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... ture of the module (2) is shown in §1, where we prove Theorem A. Let Rbe a local domain, and E a finitely generated R-module. If ap(E) is a principal ideal for all p, then £ = @pR/ap(E) (direct sum). By investigating some of the properties of «,(£), where s = ext rank £, we obtain, in §3, proofs of the following two theorems: Theorem B. Let R be a noetherian ring (3) , and E a finitely generated Rmodule. If the dimension of the R/m vector space £/m£ 75 the same for each maximal ideal m in R, and if r denotes the radical of R, then E/xE is R/xprojective.