Above we used the undefined product ***.
It is defined by *A*B = Tor_1(A,B)*, the *torsion product*
of *A* and *B*. It is zero when either *A* or *B*
is free, and we have *A*B = B*A*.

Just for completeness the definitions.
A *resolution* of *A* is an exact sequence
*...-> C_i -> ... -> C_0 -> A -> 0*.
It is called *free* when all *C_i* are free.

Free resolutions exist, and any two free resolutions are chain homotopic.
Now *Tor_i(A,B) = H_i(C tensor B)* is independent of the choice
of free resolution *C*.

(And dually, *Ext^i(A,B) = H^i(Hom(A,B))*.)

Over a principal ideal domain one finds *Tor_i(A,B) = Ext^i(A,B) = 0*
for *i > 1*. Also, *Tor_0(A,B) = A tensor B*, so the only
torsion module of interest here is *A*B*.

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