I think you are looking for semifields in this sense:
In ring theory, combinatorics, functional analysis, and theoretical computer science (MSC 16Y60), a semifield is a semiring (S,+,·) in which all elements have a multiplicative inverse.
However, I think the passage I copied contains an error when it says all elements. If you look at the references you'll find that they do exclude $0$.
You'll also find, in Golan's book at least, the name "division semiring" used if commutativity of multiplication is required.
The two most useful resources on semirings that I ever found were these:
Golan, Jonathan S. Semirings and their Applications. Springer Science & Business Media, 2013.
Gondran, Michel, and Michel Minoux. Graphs, dioids and semirings: new models and algorithms. Vol. 41. Springer Science & Business Media, 2008.
I have never read this but it looks like something to consult:
Glazek, Kazimierz. A guide to the literature on semirings and their applications in mathematics and information sciences: with complete bibliography. Springer Science & Business Media, 2002.