This thesis is an extension (and generalization) of the work(s) done by Faith ([12]) . . As a generalization of the right (left) annihilator , we introduce the notion of a weak annihilator of a subset in a ring . Weak annihilator of in is defined by set of ; which denoted by . If is singleton , say , we use in place of . We introduce Ore extension ring , and denoted by whose elements's are the polynomials over .
