This paper studies the nilpotent
ring analogues of several well-known results on finite p-groups. We first prove an
analogue for finite nilpotent p-rings [a ring is called a p-ring if its additive group is a
p-group] of the Burnside Basis Theorem, and use this to obtain some information on
the automorphism groups of these rings. Next we obtain Anzahl results,
showing that the number of subrings, right ideals, and two-sided ideals of a
given order in a finite nilpotent p-ring is congruent to 1 modp. Finally, we
characterize the class of nilpotent p-rings which have a unique subring of a given
order.
The analogy between nilpotent groups and nilpotent rings which motivates the
results of this paper is the replacement of group commutation by ring product.
A nilpotent ring, of course, is itself a group under the circle composition
x ∘ y = x + y + xy but the structure of this group implies little about the
invariants to be studied here, as shown by the examples in the last section of the
paper.
