Gaussian trivial ring extensions and fqp-rings

Francois Couchot (1)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme

Let A be a commutative ring and E a non-zero 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 infinite.

Mots clés: chain ring; Gaussian ring; arithmetical ring; Bézout ring

Domaines: Anneaux et algèbres [math.RA]; Algèbre commutative [math.AC]

Francois Couchot. Gaussian trivial ring extensions and fqp-rings. Communications in Algebra, 2015, 43 (7), pp.2863--2874. ⟨hal-00881655v2⟩