In nuclear chemistry, how does a neutron split to form a proton and an electron?
One neutron has changed into a proton, that's what has happened. Our convention in chemistry is to identify nuclear species by the proton number (or the "atomic number") and that is why a new nuclide with an increased atomic number is formed. Both the $\beta^+$ and $\beta^-$ decay follow identical paths: in $\beta^-$ decay, a neutron changes into a proton (thus giving no change in mass number and increase in atomic number), whereas in $\beta^+$ decay, a proton decays into a neutron giving no change in the mass number and a decrease in the atomic number.