Is there a definition of a determinant which can be applied to a module with no basis?
We can produce a module with noncommutative rings, without knowing a basis for these rings, i.e. without units. Can we assign an "area" value to a field extension of the entire family of rings with the unit of the extension, and without establishing a basis surrounding that unit?
Follow up question, if there is such an area-finding function of a general module, will it persist for modules $M,N$ that det$(MN)$$=$(det$M$)(det$N$) and det$(NM)$$=$(det$N$)(det($M$)?