I need to work with real numbers, but extended to have an additional element. This element, I denote by $\odot$ and my set is: $\mathbb{R}_{\odot}=\mathbb{R}\cup\{\odot\}$. This element should behave as special "tagged" zero. For example $1+\odot=1$. That said I would like to treat this set as much as possible as standard real numbers. For example it would be great to treat it as a ring and be able to re-use all ring-related theory. Are there any know ways of construcing algebraic structures like this? Any other suggestions of how to best represent something like this?
P.S. Sorry if my explanation is too informal. This is outside of my field so I am asking for help here.