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Here W is just a semigroup under ‘+’. Several interesting properties about these structures can be had from [122]. 6 Near-rings and S-near-rings In this section we introduce the concept of near-rings and Smarandache near-rings. As it is essential to know these concepts for building up binear-rings and Smarandache binear-rings, which are very new concepts, we have briefly given them. The study of Smarandache near-rings is very recent; introduced only in the year 2002 [43, 118, 126]. 1: A near-ring is a set N together with two binary operations ‘+’ and 'y' such that i.

N1 → N2 is a near-ring homomorphism; φ need not be even defined on whole of S1. For more about these please refer [118, 126]. Now the concept of Smarandache integral, equiprime and left infra near-rings are defined in an analogous and appropriate ways. These studies are solely assigned to the reader. 49: A Smarandache composition near-ring (S-composition nearring) is a quadruple (C, +, o) where (C, +, y) and C, +, y) are S-near-rings such that (a y b) o c = (a o b) y c for all a, b, c ∈ C. 50: A non zero S-ideal H of G is said to be Smarandache uniform (Suniform) if for each pair of S-ideals K1 and K2 of G such that K1 ∩ K2 = (0); K1 ⊂ H, K2 ⊂ H implies K1 = (0) or K2 = (0).

V. vi. (D, +, 0) is a loop under ‘+’. a + b = 0 implies b + a = 0 for all a, b ∈ D. (D∗, y, 1) is a group where D∗ = D\ {0}. 0 y a = a y 0 = 0 for all a ∈ D. a y (b + c) = a y b + a y c for all a, b, c ∈ D. For every pair a, b ∈ D there exists da, b ∈ D∗ such that for every x ∈ D; a + (b + x) = (a + b) + da, b x. Now [118 and 126] has defined loop near domains analogous to group rings. DEFINITION [118]: Let L be a finite loop under ‘+’ and D be a near domain, the loop near domain DL contains elements generated by di mi where di ∈ D and mi ∈ L where we admit only finite formal sums satisfying the following: i.