  We derive a $U(1)_{B-L}$-extension of the Standard Model from a generalized
Connes-Lott model with algebra ${\mathbb C}\oplus{\mathbb C}\oplus {\mathbb
H}\oplus M_3({\mathbb C})$. This generalization includes the Lorentzian
signature, the presence of a real structure, and a weakening of the order $1$
condition. In addition to the SM fields, the model contains a $Z_{B-L}'$ boson
and a complex scalar field $\sigma$ which spontaneously breaks the new
symmetry. This model is the smallest one which contains the SM fields and is
compatible with both the Connes-Lott theory and the algebraic background
framework.
