Re: sub-structures of free structures

John Stell asked when to expect subobjects of free objects to be free and observed that such is the case for modules over prinicpal ideal domains. For commutative rings that's the only case: if every ideal is free as a module then the ring is a PID. He points out that subalgebras of free algebras are free if the theory consists of just one binary operation and no equations. Isn't that the case for any equationless theory?