Abelian Groups - László Fuchs [Springer]

622 16 Endomorphism Rings
2 Endomorphism Rings of p-Groups
It is a rather intriguing question to find general properties shared by the endomorphism
rings of p-groups. Fortunately, substantial information is available, and we
wish to discuss some relevant results.
Role of Basic Subgroups We start with the observation that underscores the
relevance of basic subgroups also from the point of view of endomorphisms: any
endomorphism of a reduced p-group is completely determined by its action on a
basic subgroup. Actually, a stronger statement holds: any homomorphism of a p-
group A into a reduced group C is determined by its restriction to a basic subgroup
B of A. The exact sequence 0 ! B ! A ! A=B ! 0 induces the exact sequence
0 D Hom.A=B; C/ ! Hom.A; C/ ! Hom.B; C/ which justifies our claim.
The Finite Topology It should be pointed out that in case of reduced p-groups,
the finite topology of the endomorphism ring can be defined intrinsically, without
reference to the underlying group. If x 2 A, then there are a projection W A !hci
onto a cyclic summand and a 2 E such that x D c. Manifestly, the neighborhood
U x (annihilating x) is nothing else than the left annihilator ideal ./ ? .
Proposition 2.1. Let A be a separable p-group.
(i) The finite topology of its endomorphism ring E can be defined by taking the left
annihilators of the primitive idempotents.
(ii) In the finite topology of E, the left ideal E 0 of E generated by the primitive
idempotents is dense, and its completion is E.
Proof. As U x U px and every element is contained in a finite summand, the U for
primitive idempotents form a subbase.
For (ii) we show that for every 2 E and for every neighborhood U x ,thecoset
C U x intersects E 0 .NowU x D .p k / ? (if hxi Dp k A) is a left ideal, so 1 2 U x
implies .1 / 2 U whence 2 C U follows. ut
Structure of End for p-Groups We are in the fortunate situation that a lot is
known about the endomorphism rings of p-groups. As a matter of fact, Liebert [3]
gave a complete characterization in the separable case. A much less informative, but
perhaps more attractive information is recorded in the next theorem.
Before stating the theorem, we recall a definition: a ring E is a split extension of
a subring R by an ideal L of E if there exists a ring homomorphism W E ! R that
is the identity on R, andL D Ker. We write E D R ˚ L (direct sum in the group
sense).
Theorem 2.2 (Pierce [1]). For a p-group A, End A is a split extension
End A Š R ˚ End s A;
where R is a ring whose additive group is the completion of a free p-adic module,
and End s A is the ideal of small endomorphisms of A.