  
  [1XB The Mathematical Idea behind [5Xhomalg[1X[0X
  
  As  finite  dimensional  constructions in linear algebra over a field k boil
  down  to  solving  (in)homogeneous  linear  systems  over  k,  the  Gaussian
  algorithm makes the whole theory perfectly computable.
  
  Hence, for homological algebra (viewed as linear algebra over general rings)
  to be constructive one needs to find apropriate substitutes for the Gaussian
  algorithm,  where  finite  dimensionality  has  to  be  replaced  by  finite
  generatedness.
  
  Luckily  such  substitutes  exist  for  many  rings  of interest. Beside the
  well-known  Hermite normal form algorithm for principal ideal rings it turns
  out that apropriate generalizations of the classical Gröbner basis algorithm
  for  polynomial  rings  provide  the  desired substitute for a wide class of
  commutative [13Xand[0m noncommutative rings. Note that for noncommutative rings the
  above  discussion  has to be restricted to homological constructions leading
  to one-sided linear systems XA=B resp. AX=B (--> [14X1.1-5[0m).
  
  The  two  appendices [14XC[0m and [14XD[0m provide the list of matrix operations needed by
  [5Xhomalg[0m  to  perform  homological computations. Subsection [14X1.1-3[0m explains how
  these matrix operations can be delegated to external systems.
  
