For inverting in the original parent ("internally"), the name
"m.inverse_of_unit()" is used widely in Sage, e.g. in integer rings,
polynomial rings, p-adic rings, etc. The name "m._invert_unit()" seems to
be used only for matrices. I now understand the "unit" in the name for
differentiating from the inverse involving change of parent by coercion.
So I vote for "inverse_of_unit" for standardization.
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