GLOBAL SUBDIRECT PRODUCTS
A celebrated theorem of Birkhoff asserts that every (non-trivial)
algebra is isomorphic to a subdirect product of subdirectly irreducible
algebras. Why is this not the ultimate representation theorem of (universal)
algebra? There are two reasons for this. The basic idea behind represen-
tation theorems is to "decompose" a given structure into "simpler" structures
in such a way that the properties of the given structure can be "reduced" to
the properties of the simpler structures. In the case of Birkhofffs Theorem
the "simpler" structures are subdirectly irreducible and the "decomposition"
is subdirect. Now, first of all it is usually very difficult to determine the
subdirectly irreducible factors (that is the subdirectly irreducible homo-
morphic images) of a given algebra. In fact, frequently these factors are
just not known at all (as is the case,for example, for arbitrary groups or
rings). However, there are important cases where the subdirect factors are
known. For example, an abelian group is subdirectly irreducible if and only if
it is cocyclic (that is quasi cyclic or cyclic of prime power order). Thus we
obtain: Every (non-trivial) abelian group is isomorphic to a subdirect pro-
duct of cocyclic groups. Although the properties of cocyclic groups are very
well known, this theorem plays practically no role in abelian group theory.
This is due to the second weakness of
Birkhofffs
Theorem: Subdirect products
in general are so "loose" that very little can be inferred from the proper-
ties of the factors. In the structure theory of abelian groups the central
role is played by direct sum representations. Direct sums are special sub-
direct products which are "tight" enough so that significant information can
be obtained from the properties of the factors.
To further illustrate this weakness of Birkhoffs Theorem, let us con-
sider another subdirect representation by subdirectly irreducibles : A (non-
trivial) commutative ring is semi-simple (that is has trivial Jacobson
radical) if and only if it is a subdirect product of fields. Although this is
an important result of ring theory (in fact it motivated the definition of
Received by the Editors July 27,1977.
The second author was partially supported for this work at the Gesamthoch-
schule Kassel by a grant from the Alexander von Humboldt Foundation of
West Germany.
1
Previous Page Next Page