What type of nuclear decay results in the atomic number increasing by one with no change in mass?

There are a variety of nuclear decay reactions, including alpha decay, beta decay and gamma decay reactions. In alpha decay, an alpha particle (Helium atom, with a mass number of 4 and atomic number of 2) is released. In beta decay reaction, an electron is released. An electron is considered a particle with negligible mass and a charge of -1. When the electron is released, the charge of the atom increases by 1 and hence its mass number stays the same, but the atomic number increases by 1. Here is an example,

`C^14 _6 -> N^14 _7 + e^0 _-1`

Here, Carbon-14 undergoes beta decay and nitrogen is generated. 

In gamma decay reactions, a gamma particle (does not contain mass or charge) is released and hence there is no change in atomic number or mass number of the atom.

Hope this helps.