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homalg 
definitions of the basic GAP4 categories 
and some tool functions (e.g. homalgMode ) 

homalgTable 
dictionaries between MatricesForHomalg 
and the computing engines  
HomalgRing 
internal and external rings 
HomalgRingMap 
ring maps 
HomalgMatrix 
internal and external matrices 
HomalgRingRelations 
a set of ring relations 
In the following CAS or CASystem mean computer algebra systems.
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Content 
Tools 
the elementary matrix operations that can be 
overwritten using the homalgTable  
(and hence delegable even to other CASystems)  
Service 
the three operations: basis, reduction, and syzygies; 
they can also be overwritten using the homalgTable  
(and hence delegable even to other CASystems)  
Basic 
higher level operations for matrices 
(cannot be overwritten using the homalgTable) 
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Content 
LIRNG 
logical implications for rings 
LIMAP 
logical implications for ring maps 
LIMAT 
logical implications for matrices 
COLEM 
clever operations for lazy evaluated matrices 
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ResidueClassRingForHomalg 
some global variables 
ResidueClassRing 
residue class rings, their elements, and matrices, 
together with their constructors and operations  
ResidueClassRingTools 
the elementary matrix operations for matrices 
over residue class rings  
ResidueClassRingBasic 
the three operations: basis, reduction, and syzygies 
for matrices over residue class rings  
For the purposes of homalg, the ring of integers is, at least up till now, the only ring which is properly supported in GAP4. The GAP4 builtin cababilities for polynomial rings (also univariate) and group rings do not statisfy the minimum requirements of homalg. The GAP4 package Gauss enables GAP to fullfil the homalg requirements for prime fields, and ℤ / p^n.
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Integers 
the homalgTable for the ring of integers 
generated by GAPDoc2HTML