On the subring structure of finite nilpotent rings

Abstract

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 mod p. 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 o 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. All rings considered here are associative. The reader may verify, however, that all results of §§ 1-3 hold without the assumption of associativity, with the exception of (3.3). The unqualified word means two-sided ideal. The letter p always denotes a prime number. If 31 is a ring, we denote the additive group of 9t by 3ϊ +. The order of a ring 3ΐ, denoted |9t|, is the order of the group 9ϊ+; the index of a subring @ in a ring 3ΐ, denoted [9t: @], is the index of Θ + in 3ϊ +. A ring is called null if all products are 0. A ring 9t is called nilpotent of exponent e if all products of e elements from 9ΐ are 0, but not all products of e — 1 elements are 0. The characteristic of a finite ring is the maximum of the additive orders of its elements. The smallest ideal containing ideals @ and % is denoted @ + %. We shall need the following elementary results: