  
  [1XA [33X[0;0YThe Mathematical Idea behind [5XModules[105X[101X[1X[133X[101X
  
  [33X[0;0YAs  finite  dimensional  constructions in linear algebra over a field [22Xk[122X boil
  down  to  solving  (in)homogeneous  linear  systems  over  [22Xk[122X,  the  Gaussian
  algorithm makes the whole theory perfectly computable.[133X
  
  [33X[0;0YHence, for homological algebra (viewed as linear algebra over general rings)
  to  be computable one needs to find appropriate substitutes for the Gaussian
  algorithm,  where  finite  dimensionality  has  to  be  replaced  by  finite
  generatedness.[133X
  
  [33X[0;0YLuckily  such  substitutes  exist  for  many  rings  of interest. Beside the
  well-known  Hermite normal form algorithm for principal ideal rings it turns
  out   that  appropriate  generalizations  of  the  classical  Gröbner  basis
  algorithm  for  polynomial  rings  provide the desired substitute for a wide
  class  of commutative [13Xand[113X noncommutative rings. Note that for noncommutative
  rings the above discussion has to be restricted to homological constructions
  leading to one-sided linear systems [22XXA=B[122X resp. [22XAX=B[122X (--> [14X1.1-3[114X).[133X
  
