  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 B{\'e}zout 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.
