Gaussian Trivial Ring Extensions and FQP-rings

Abstract

Let A be a commutative ring and E a nonzero A-module. Necessary and sufficient conditions are given for the trivial ring extension R of A by E to be either arithmetical or Gaussian. The possibility for R to be Bézout is also studied, but a response is only given in the case where pSpec(A) (a quotient space of Spec(A)) is totally disconnected. Trivial ring extensions which are fqp-rings are characterized only in the local case. To get a general result we intoduce the class of fqf-rings satisfying a weaker property than fqp-ring. Moreover, it is proven that the finitistic weak dimension of a fqf-ring is 0, 1 or 2 and its global weak dimension is 0, 1, or ∞.

Key Words:

• Arithmetical ring
• Bézout ring
• Chain ring
• Gaussian ring

2010 Mathematics Subject Classification:

• 13F30
• 13C11
• 13E05

Notes

¹we prefer “chain ring” to “valuation ring” to avoid confusion with “Manis valuation ring.”

Communicated by S. Bazzonio.