Smarandache Near-rings

Generally, in any human field, a Smarandache Structure on a set A means a weak structure W on A such that there exists a proper subset B in A which is embedded with a stronger structure S.
These types of structures occur in our everyday life, that’s why we study them in this book.
Thus, as a particular case:
A Near-Ring is a non-empty set N together with two binary operations ‘+’ and ‘.’ such that (N, +) is a group (not necessarily abelian), (N, .) is a semigroup. For all a, b, c in N we have (a + b) . c = a . c + b . c.
A Near-Field is a non-empty set P together with two binary operations ‘+’ and ‘.’ such that (P, +) is a group (not necessarily abelian), (P \ {0}, .) is a group. For all a, b, c in P we have (a + b) . c = a . c + b . c.
A Smarandache Near-ring is a near-ring N which has a proper subset P in N, where P is a near-field (with respect to the same binary operations on N).