Top exterior powers over commutative rings

Document Type

Article

Publication Date

1-1-1999

Abstract

We recall that the class of FGC rings are those rings for which every finitely generated R-module may be written as a direct sum of cyclic R-modules and demonstrate that it is coincident with a new class of rings. When the top exterior power of an R-module M is cyclic, we define the non-negative integer p = crank(M), and show that under the hypothesis that the annihilator of the top exterior power is contained in the Jacobson radical, it counts the maximum number of times that the top exterior power can appear as a direct summand of M. If P is finitely generated projective of constant rank n, then an isomorphism Hom(P, ∧n P) ≅ ∧n-1 P is set up. Finally, a unimodular condition is shown to be a consequence whenever right multiplication is a split map onto the top exterior power of P. © 1999 OPA (Overseas Publishers Association) N.V. Published by license under the Gordon and Breach Science Publishers imprint.

Publication Title

Linear and Multilinear Algebra

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