Let R be an arbitrary unital
commutative ring and A a unital R-algebra, i.e.\
a unital ring A together with a ring homomorphism from R into the center
of A sending 1 to 1.
We denote by 
the category of finitely generated R-modules
and by 
the category of right A-modules which are finitely
generated as R-modules. Assume that the isomorphism classes
of and form sets 
and 
. The
monoidal structure of the category - coming along with the
tensorproduct over R - induces the structure of a
commutative monoid on
. Thus the multiplication of two elements 
given by representatives 
is defined as the isomorphism
class of 
. The unit element 1 is the isomorphism class of
R. and in addition you have an element 0 corresponding to the zero module.
In a similar manner there is an action of the monoid from
the left on the set , since the tensor product
of an R-module U with a right A-module gives
again a right A-module.
Now let 
be the monoid algebra on over
the integers and 
the free abelian group on .
The above action of on induces a 
-module
structure on . Let 
be the abelian subgroup of
generated be the expressions U'-U+U'' for all 
, such that there is a short
split exact sequence (from now on we will not deitinguish between
elements in resp. and representatives for these
elements in resp. ).
Since tensoring split exact sequences yields again split exact
sequences, is obviously an ideal in . We denote the
residue class ring by
Similarily,
in we consider the abelian subgroup 
generated by all
expressions M'-M+M'', such that there is a short exact sequence
of right A-modules which is split as a sequence of R-modules.
It is obviously an -submodule of and clearly one
has
Therefore the quotient
can be considered as an 
-module. Before proceeding let us
give some examples. For this purpose denote by 
the set of isomorphism classes corresponding to the
indecomposable R-modules and by 
the
set corresponding to cyclic R-modules, i.e. modules
isomorphic to R/I for some ideal I in R. Note that 
is a submonoid of and both contain 1 and 0.
Therefore, the 
-linear span of in is a
subring.
Now if R is noetherian, then the ring is generated by the
residue classes of 
. If in addition the Krull-Schmidt
theorem holds in , that is, if decomposition into
indecomposables is unique, then is free abelian on
.
Let us look in more detail to the case where R
is a principal ideal domain. Here you have 
and the former is a submonoid as well. More precisely, for each pair
(p,l) of a prime 
and a positive integer 
you have
an element in whose representative is 
and which
we denote by 
. From the main theorem on modules over principal
ideal domains it follows that for different pairs 
you get different elements 
and that all elements
of - except 1 and 0 - are of this form. The multiplication
in for elements other than 1 and 0
is given by the rule
Now is canonically isomorphic to
the integral monoid algebra of factored by the one
dimensional span of the element 
. In the case
R is a principal ideal domain we will always identify in
this way and use the notation for basis elements as above.
In the same manner we write
[n] for the residue class of the isomorphism type of R/(n) if
is arbitrary. Thus 
if 
is the
primedecomposition of n in R.
In the case where R is a field, is canonically isomorphic
to and 
is the
Grothendieck group 
of A considered as a -module.
We call a right A-module 
reducible by R-sums
iff M contains an A-submodule N which is a direct summand
of R-modules in M. If M is not reducible by R sums we call it
irreducible by R sums. The concept is clearly invariant under
isomorphisms of A-modules.
Let 
be the subset of consisting of
isomorphism classes of by R sums
irreducible A-modules. If R is noetherian it is easily seen
that is generated by as a -module.
