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.

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