In this thesis, we introduce the set S(R) of “ strong zero-divisors” in a ring R and extend some useful and well known results on zero-divisors of commutative rings to non-commutative ones. For example, it is shown that for a ring R, 1 |S(R)| ¥, if and only if R is finite and not a prime ring. When certain sets of ideals have ACC, we show that either S(R) =R or S(R) is a union of prime ideals each of which is a left or a right annihilator of a cyclic ideal. Also, for any ring R, we associate an undirected graph, say (R), and investigate the interplay between the ring-theoretic properties of R and the graph-theoretic properties of (R). Let S be a commutative ring with identity and M be a left S-module. Then we express a number of necessary and sufficient condition for the multiplication “.” o a module M with a finite quasi S-basis A such that (M,+,.)’s become associative S-algebras. In particular, we give a number of necessary and sufficient condition for the multiplications introduced on