On the Number of Smarandache Zero-Divisors and Smarandache Weak Zero-Divisors in Loop Rings‪.‬

Abstract In this paper we find the number of smarandache zero divisors (S-zero divisors) and smarandache weak zero divisors (S-weak zero divisors) for the loop rings [Z.sub.2][L.sub.n](m) of the loops [L.sub.n] (m) over [Z.sub.2]. We obtain the exact number of S-zero divisors and S-weak zero divisors when n = [p.sup.2] or [p.sup.3] or pq where p, q are odd primes. We also prove Z[L.sub.n](m) has infinitely many S-zero divisors and S-weak zero divisors, where Z is the ring of integers. For any loop L we give conditions on L so that the loop ring [Z.sub.2]L has S-zero divisors and S-weak zero divisors. [section] 0. Introduction