Thesis (Ph.D.) - Cairo University - Faculty of Science - Department of Mathematics

Abstract affine near-rings form an interesting variety of abelian near-rings. We prove that there is a category equivalence between the category of abstract affine near-rings and the category of modules over unspecied rings. Several properties of this equivalence are given. As applications, one can easily transfer first order properties of modules (with respect to the two-sorted first order language of modules) to the corresponding properties of abstract affine near-rings (with respect to the one-sorted first order language of near-rings). With the aid of a Theorem of Gonshor, we study the relation between different kinds of ideals of an arbitrary abstract affine near-ring A and those of the matrix near-ring Mn (A); n{u2265}1: We prove that there is a one-to one correspondence between the r-ideals (resp. pure r-ideals , strongly pure r-ideals) of A and those of Mn(A). Semirings occur in different mathematical fields and have also become of great interest as a tool in different branches of computer science

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