M is torsion-free because it does not contain any elements that can be multiplied by a non-zero element in the ring R to equal zero. In other words, there are no elements in M that have a non-trivial annihilator in R.
M is rank 1 because it can be generated by a single element. This means that there is a single element in M that can generate all other elements through multiplication by elements in the ring R.
M is not a free R-module because it does not have a basis. In other words, there is no set of elements in M that can generate all other elements through linear combinations with coefficients in R.
Yes, M may be a free module over a different ring. The definition of a free module depends on the ring, so M may satisfy the conditions for being a free module over one ring but not over another.
One example is the module of all rational numbers over the ring of integers. Another example is the module of all real numbers over the ring of continuous functions on the interval [0,1]. In both cases, the modules are rank 1 because they can be generated by a single element, but they are not free because they do not have a basis.