# Smarandache Near-Rings

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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 I 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).96. 97. 98. 99. 100. 101. 102. 103. VASANTHA KANDASAMY, W. B., The units of semigroup seminear-rings, Opscula Math, ... gallup. unm. edu/~smarandache/V as antha-Book2 . pdf VASANTHA KANDASAMY, W. B., Smarandache near-rings, anbsp;...

 Title : Smarandache Near-Rings Author : W. B. Vasantha Kandasamy Publisher : Infinite Study - 2002