|  7.4.2 Groebner bases in G-algebras 
We follow the notations, used in the SINGULAR Manual (e.g. in  Standard bases).
 
For a  -algebra  , we denote by  the left submodule of a free module  , generated by elements  . 
Let  be a fixed monomial well-ordering on the  -algebra  with the Poincar@'e-Birkhoff-Witt (PBW) basis  .
For a given free module  with the basis  ,  denotes also a
fixed module ordering on the set of monomials  . 
  Definition For a set , define  to be the  -vector space, spanned on the leading monomials
of elements of  ,  . 
We call  the span of leading monomials of  . 
Let  be a left  -submodule.
A finite set  is called a left  Groebner basis of  if and
only if  , that is for any  there exists a  satisfying  , i.e., if  , then  with  . 
Remark: In general non-commutative algorithms are working with global well-orderings
only (see  PLURAL,  Monomial orderings and  Term orderings), unless we deal with
graded commutative algebras via  Graded commutative algebras (SCA).
 
A Groebner basis  is called minimal (or reduced) if  and if  for all  .
Note, that any Groebner basis can be made minimal by deleting successively those  with  for some  . 
For  and  we say that  is completely reduced with
respect to  if no monomial of  is contained in  . 
  Left Normal Form 
A map 
 , is called a (left) normal form
on  if for any  and any left Groebner basis  the following
holds: 
(i) 
 , 
(ii)  if 
 then  does not divide  for all  , 
(iii) 
 . 
 is called a left normal form of  with
respect to  (note that such a map is not unique). 
Remark:
As we have already mentioned in the definitions
 idealandmodule(see
  PLURAL), byNF(orreduce) PLURAL understands a left normal form.
 Note, thatrightNFfrom  nctools_lib allows to compute a right normal form. 
  Left ideal membership (plural) 
For a left Groebner basis  of  the following holds:  if and only if the left normal form  . 
For computing a left Groebner basis GofI, use  std (plural). 
For computing a left normal form of fwith respect toG, use  reduce (plural). 
  Right ideal membership (plural) 
The right ideal membership is analogous to the left one:
 
for computing a right Groebner basis GofI, use  rightStd from  nctools_lib, 
for computing a right normal form of fwith respect toG, use  rightNF from  nctools_lib. 
  Two-sided ideal membership (plural) 
Let  be a two-sided ideal and  be a two-sided Groebner basis of  . 
Then  if and only if the left normal form  . 
For computing a two-sided Groebner basis TofJ, use  twostd (plural), 
for computing a normal form of fwith respect toT, use  reduce (plural). 
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