Some remarks on surjections of unit groups

The main purpose of this paper is to study unital ring homomorphisms of associative rings $$$\varphi : R\rightarrow S$$$ satisfying one of the following conditions: (a) the unit-preserving property: $$$\varphi(R^{\times})=S^{\times}$$$ and (b) the inverse unit-preserving property: $$$\varphi^{-1}(S^{\times})=R^{\times}$$$.
We establish the relationship between these two conditions. Several characterizations of such conditions are settled. An application to the index of unit groups of rings $$$R\subset S$$$ having a nonzero common ideal is given.

The main purpose of this paper is to study unital ring homomorphisms of associative rings $$$\varphi : R\rightarrow S$$$ satisfying one of the following conditions : (a) the unit-preserving property : $$$\varphi(R^{\times})=S^{\times}$$$ and (b) the inverse unit-preserving property : $$$\varphi^{-1}(S^{\times})=R^{\times}$$$.
We establish the relationship between these two conditions. Several characterizations of such conditions are settled. An application to the index of unit groups of rings $$$R\subset S$$$ having a nonzero common ideal is given.

Associative ring ideal group of units left Artinian von Neumann regular Jacobson radical

Associative ring ideal group of units left Artinian von Neumann regular Jacobson radical