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Local-Global Criteria for Outer Product Rings

Published online by Cambridge University Press:  20 November 2018

Dennis R. Estes  and
Jacob R. Matijevic
Dennis R. Estes
Affiliation:
Queen's University, Kingston, Ontario
Jacob R. Matijevic
Affiliation:
Queen's University, Kingston, Ontario
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Let R be a commutative ring with multiplicative identity. We say that R is an outer product ring (OP-ring) if each vector v in the exterior power ⋀nRn+1 is decomposable; i.e. v = v1 ⋀ v2 ⋀ … ⋀ vn with vi ∈ Rn+1 (alternatively, each n + 1 tuple of elements in R is the tuple of n × n minors of some n × (n + 1) matrix with entries in R).

Lissner initiated the study of which commutative rings are OP-rings, showing that any Dedekind domain is an OP-ring [13]. Towber classified the local Noetherian OP-rings as those with maximal ideal generated by two elements, and Hinohara's reformulation of Towber's result extended the context to semi-local rings [18, 11]. Assuming the stronger condition that all Pliicker vectors are decomposable and designating such rings as Towber rings, Lissner and Geramita gave necessary conditions for a Noetherian ring to be a Towber ring [14].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 6 , 01 December 1980 , pp. 1353 - 1360
Copyright
Copyright © Canadian Mathematical Society 1980

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