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 Rn≠0 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.
