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Is there accepted terminology for algebraic structures whose every subalgebra is free?

Examples:

  • Any free group
  • Any vector space
  • More generally, any free module over a PID. In fact, this characterizes PID's; given a commutative unital ring $R$, every free $R$-module enjoys the property of interest iff $R$ is a PID.
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    $\begingroup$ You might be interested in this mathoverflow thread. It's a different question, but related. $\endgroup$ – Alex Kruckman Oct 5 '14 at 7:34
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    $\begingroup$ One might say "hereditarily free". $\endgroup$ – Zhen Lin Oct 5 '14 at 8:42

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