Some Studies Of Weakly Nil-symmetric Rings

A ring R is called weakly nil-symmetric if abc =0 implies acb=0 for each a∈N(R)and b,c∈R.Weakly nil-symmetric rings are proper generalization of symmetric rings.In the second chapter of this Master’s thesis,we study some properties of weakly nil-symmetric rings and the relations among related rings.Mainly,we prove the following results:(1)R is a reduced ring if and only if the 2×2 upper triangular matrix ring T2(R)is a weakly nil-symmetric ring.(2)R is a reduced ring if and only if the 3×3 special upper triangular matrix ring V3(R)is a weakly nil-symmetric ring.(3)If e is a central idempotent of R,then R is a weakly nil-symmetric ring if and only eRe and(1-e)R(1-e)are all weakly nil-symmetric rings.(4)Let I be a reduced ideal of R.If R/I is a weakly nil-symmetric ring,then so is R.In the third chapter,we discuss the strong regularity of weakly nil-symmetric rings as follows:(5)If R is a weakly nil-symmetric ring and a∈R with a ∈ aRa,then a∈a2R∩Ra2.(6)Weakly nil-symmetric left SF rings are strongly regular.(7)Weakly nil-symmetric left MVNR rings are strongly regular.In the fourth chapter,we investigate some properties of left min-abel rings as follows:(8)Weakly nil-symmetric rings are left min-abel.(9)R is a left min-abel ring if and only if for each a∈N(R),b∈R and e∈ME1(R),abe = 0 implies aeb = 0.(10)Weakly nil-symmetric rings are 2-primal.In the fifth chapter,we present some properties of weakly nil-symmetric exchange rings,such as we prove that weakly nil-symmetric exchange rings are left quasi-duo and clean.