On the hearts of subdirectly irreducible near-rings (Q1907283)

It is well known that the heart of a subdirectly irreducible ring is either a simple ring or square zero. The situation is radically different for near-rings. The authors illustrate this with the following result: Given any prime number \(p\), there exists a zero-symmetric near-ring \(N\) with a chain of subnear-rings \(N=N_0\supset N_1\supset\cdots\supset N_p\supset\dots\) such that: (a) \((N,+)\) is a direct sum of countably many cyclic groups of order \(p\); (b) \(N_i=N^{2^i}\) for all \(i=1,2,\dots\); (c) \(N_i\) is a subdirectly irreducible near-ring with heart \(N_{i+1}\) for all \(i=0,1,\dots\); (d) \(\bigcap^\infty_{i=1} N_i=0\).