Vol. 30, No. 3, 1969

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Zero square rings

Richard Peter Stanley

Vol. 30 (1969), No. 3, 811–824

A ring R for which x2 = 0 for all x R is called a zero-square ring. Zero-square rings are easily seen to be locally nilpotent. This leads to two problems: (1) constructing finitely generated zero-square rings with large index of nilpotence, and (2) investigating the structure of finitely generated zerosquare rings with given index of nilpotence. For the first problem we construct a class of zero-square rings, called free zero-square rings, whose index of nilpotence can be arbitrarily large. We show that every zero-square ring whose generators have (additive) orders dividing the orders of the generators of some free zero-square ring is a homomorphic image of the free ring. For the second problem, we assume Rn0 and obtain conditions on the additive group R+ of R (and thus also on the order of R). When n = 2, we completely characterize R+. When n > 3 we obtain the smallest possible number of generators of R+, and the smallest number of generators of order 2 in a minimal set of generators. We also determine the possible orders of R.

Mathematical Subject Classification
Primary: 16.32
Received: 9 September 1968
Published: 1 September 1969
Richard Peter Stanley