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It's called an "almost Dedekind domain" in the literature on non-Noetherian commutative algebra. Every almost Dedekind domain is a Pr"ufer domain, or equivalently, locally a valuation domain. However, there exist Pr"ufer domains that are not almost Dedekind, e.g. any valuation domain that's not a PID. Both classes of domains come up a lot in the literature.

http://www.ams.org/journals/proc/1964-015-05/S0002-9939-1964-0166212-8/S0002-9939-1964-0166212-8.pdf

It's called an "almost Dedekind domain" in the literature on non-Noetherian commutative algebra. Every almost Dedekind domain is a Prüfer domain, or equivalently, locally a valuation domain. However, there exist Prüfer domains that are not almost Dedekind, e.g. any valuation domain that's not a PID. Both classes of domains come up a lot in the literature.

http://www.ams.org/journals/proc/1964-015-05/S0002-9939-1964-0166212-8/S0002-9939-1964-0166212-8.pdf

It's called an "almost Dedekind domain" in the literature on non-Noetherian commutative algebra. Every almost Dedekind domain is locally a valuation domain, or equivalently, a Pr"ufer domain. However, there exist Pr"ufer domains that are not almost Dedekind, e.g. any valuation domain that's not a PID. Both classes of domains come up a lot in the literature.

http://www.ams.org/journals/proc/1964-015-05/S0002-9939-1964-0166212-8/S0002-9939-1964-0166212-8.pdf

It's called an "almost Dedekind domain" in the literature on non-Noetherian commutative algebra. Every almost Dedekind domain is a Pr"ufer domain, or equivalently, locally a valuation domain. However, there exist Pr"ufer domains that are not almost Dedekind, e.g. any valuation domain that's not a PID. Both classes of domains come up a lot in the literature.

http://www.ams.org/journals/proc/1964-015-05/S0002-9939-1964-0166212-8/S0002-9939-1964-0166212-8.pdf

It's called an "almost Dedekind domain" in the literature on non-Noetherian commutative algebra. Every almost Dedekind domain is locally a valuation domain, or equivalently, a Pr"ufer domain. However, there exist Pr"ufer domains that are not almost Dedekind.

http://www.ams.org/journals/proc/1964-015-05/S0002-9939-1964-0166212-8/S0002-9939-1964-0166212-8.pdf

It's called an "almost Dedekind domain" in the literature on non-Noetherian commutative algebra. Every almost Dedekind domain is locally a valuation domain, or equivalently, a Pr"ufer domain. However, there exist Pr"ufer domains that are not almost Dedekind, e.g. any valuation domain that's not a PID. Both classes of domains come up a lot in the literature.

http://www.ams.org/journals/proc/1964-015-05/S0002-9939-1964-0166212-8/S0002-9939-1964-0166212-8.pdf