Abstract
We study the structure of $0$-primitive near-rings and are able to answer an open question in the theory of minimal ideals in near-rings to the negative, namely if the heart of a zero symmetric subdirectly irreducible near-ring is subdirectly irreducible again. Also, we will be able to classify when a simple near-ring with an identity and containing a minimal left ideal is a Jacobson radical near-ring. Such near-rings are known to exist but have unusual properties. Along the way we prove results on minimal ideals and left ideals in near-rings which so far were known to hold or have been established in the DCCN case, only.
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