Computational Commutative Algebra - Kreuzer and Robbiano
0.1 What Is This Bo ok Ab out?
The title of this b o ok is “Computational Commutative Algebra 1”. In otherwords, it treats that part of commutative algebra which is suitable for explicit
computer calculations. Or, if you prefer, the topic is that part of computer
algebra which deals with commutative ob jects like rings and mo dules. Or, as
one colleague put it jokingly, the topic could b e called “computative algebra”.
This description immediately leads us to another question. What is com-
mutative algebra? It is the study of that area of algebra in which the imp or-
tant op erations are commutative, particularly commutative rings and mo d-
ules over them. We shall assume throughout the b o ok that the reader has
some elementary knowledge of algebra: the kinds of ob jects one studies should
b e familiar (groups, rings, fields, etc.), as should some of the basic construc-
tions (homomorphisms, residue class rings, etc.). The commutative algebra
part of this b o ok is the treatment of p olynomials in one or more indetermi-
nates. To put this in a more fancy way, we could say that the generality we
shall b e able to deal with is the theory of finitely generated mo dules over
finitely generated algebras over a field.
This leaves us with one last unexplained part of the title. What do es the
“1” refer to? You guessed it! There will b e a second volume called “Com-
putational Commutative Algebra 2”. In the course of writing this b o ok, we
found that it was imp ossible to concentrate all the material we had planned
in one volume. Thus, in the (hop efully) not so distant future we will b e back
with more. Meanwhile, we suggest you get acquainted with the next 300 or
so pages, and we are confident that this will keep you busy for a while
