Suppose I want to make the following argument: Let $(A,m)$ be a local commutative ring. Then $(A,m)$ is a quotient of a local ring $(B,n)$ which is a domain, since there is a "freest local ring generated by it," which is a domain. (In my particular situation I want to reduce a theorem I know to be true about local rings to a statement about domains - the Krull intersection theorem.)
Questions:
- What is the correct terminology for what I am looking for? (Free local ring on (A,m)?)
- Does it exist? How does one construct it? (I want to say, take the free $\mathbb{Z}$-algebra generated by elements of the local ring, modulo the ideal generated by the relationship that those letters corresponding to elements that were units in A remain units. Then there is an evaluation map to the original local ring, which is a surjection. It is not completely clear to me that this is a local ring - I can reduce this to the question of whether all words that evaluate to 1 are units though.)
- Does this fit under some general construction (a solution to problems of the form: given one example of an algebraic object of some type, construct the smallest and freest object of that same class surjecting onto it.)
