In this work we show that the world of ring theory strongly bids its admirers and serious students to embark upon a conscientious and detailed adventure into the world of near-ring. As a result, one sees that the ideas and structures of each provide insight and influence upon the other. Here our main task would be to study near-rings and near-ring groups with various types of chain conditions on annihilators together with finite spanning dimensions and finite Goldie dimensions character. The reader will find that the notion of finite spanning dimensional (fsd) modules over a ring extends to near-ring groups. Here we obtain a supplement primary decomposition of the zero of an fsd near ring group. Moreover we notice a visible difference between a right Goldie near ring and a left Goldie near ring. We exhibit some extension of what is already available dealing with the right Goldie near rings thereby extending some results in this manner. An important and interesting investigation is carried out on near-rings with fgd satisfying the acc on annihilators with parts having minimum condition.