| Commutative rings is an important class of rings. To extend results on commutative rings to general rings, several generalizations of commutativity have been studied. In 1999, Cohn introduced the notion of a reversible ring.Definition 1.1 A ring R is called reversible if ab=0 implies ba=0 for a,b∈R.Anderson and Camillo studied the rings whose zero products commute, and used the term ZC2 to stand for reversible rings, while Krempa andNiewieczerzal took term C0 for it.Definition 1.2 A ring R is called symmetric if rst=0 implies rts=0 for all r,s,t∈R.Anderson and Camillo took the term ZC3 for symmetric rings. A ring Ris symmetric if and only if r1, r2,…,rn∈R , r1r2…rn=0 , with n anypositive integer, implies rσ(1)rσ(2)…rσ(n)=0 for any permutationσof the set{1,2,…,n}.Semicommutative rings is first studied by Shin.Definition 1.3 A ring R is called semicommutative if ab=0 implies aRb=0 for all a,b∈R .A ring is semicommutative if and only if any left (right) annihilator in R is an ideal of R .Definition 1.4 A ring R is called weak symmetric if rst=0 implies rts is nilpotent elements of R for all r,s,t∈R. Definition 1.5 A ring R is called weak reversible if for all a,b,r∈R such that ab=0, Rbra is a nil left ideal of R (equivalently, braR is nil right ideal of R).Clearly symmetric rings are weak symmetric, and reversible rings are weak reversible. However, weak symmetric rings are not necessarily symmetric; weak reversible rings are not necessarily reversible. Therefore weak symmetric rings are proper generalizations of symmetric rings; weak reversible rings are proper generalizations of reversible rings.Rege and Chhawchharia (1997) introduced the notion of an Armendariz ring.Definition 1.6 A ring R is said to be Armendariz if wheneverpolynomials f(x)=sum from i=0 to n aixi, g(x)=sum from j=0 to m bjxj∈R[x], satisfy f(x)g(x)=0,thenaibj=0 for all 1≤i≤n ,1≤j≤m.The name "Armendariz ring " was chosen because Armendariz in 1974, had noted that a reduced ring satisfies the condition of Armendariz ring.Definition 1.7 A ring R is said to be weak Armendariz if wheneverpolynomials f(x)=sum from i=0 to n aixi, g(x)=sum from j=0 to m bjxj∈R[x], satisfy f(x)g(x)=0, thenaibj∈Nil(R)for all 1≤i≤n ,1≤j≤m; where Nil(R) denotes the set ofnilpotent elemments of R.In this thesis, we introduce and study weakly semicommutative rings, which are generalizations of semicommutative rings.Definition 2.1 A ring R is called weakly semicommutative if ab=0 implies aRb(?)Nil(R) for all a,b∈R.The relationship between the rings mentioned above can be illustrated as follows. where wsc means weakly semicommutative, sc means semicommutative, and wrev means weakly reversible.Clearly, semicommutative rings are weakly semicommutative rings. Thus weakly semicommutative rings are a generalization of semicommutative rings. In this thesis, we study the structures of weakly semicommutative rings, matrix rings and polynomial rings over weakly semicommutative rings, and relationship of between weakly semicommutative rings, semicommutative rings, and weak Armendariz rings.The main results in this thesis are as follows.Theorem 2.2 The following statements are equivalent for a ring R.(1) R is weakly semicommutative;(2) a2=0 implies aR(?)Nil(R) for all a∈R;(3) ab=0 implies baR(?)Nil(R) for all a,b∈R.(4) R is weak reversible.Example 2.11 Let F be a field, F〈X,Y〉the free algebra on X,Y overF and I denote the ideal〈X2〉2 of F〈X,Y〉, where〈X2〉is the ideal ofF〈X,Y〉generated by X2. Consider the ring R=F〈X,Y〉/I. Then we haveNil(R)=xRx+Rx2R+Fx and N2(R)=Rx2R, where x=X+I∈R , N2(R)denotes the set of elements a∈R such that a2=0. Then R is a weakly semicommutative ring and Nil(R) is not an ideal of R.Theorem 2.14 For an I-ring R, the following statements are equivalent.(1) R is weakly semicommutative;(2) R/J(R) is reduced; (3) Nil(R)=J(R);(4) Nil(R)is an ideal of R.Theorem 2.20 If R has a reduced ideal I such that R/I is weakly semicommutative, then R is weakly semicommutative.Example 3.2 For a field F, the 2×2 matrix ring M2(F) is not weaklysemicommutative.Theorem 3.3 If M2(R) is weakly semicommutative, then R is nil.Proposition 3.5 Let F be an algebraically closed field. Then a subring S of M2(F) is weakly semicommutative if and only if either S is reduced orS is isomorphic to a subring of T2(F).Theorem 3.6 Let F be an algebraically closed field. Then a subalgebra S of Mn (F) is weakly semicommutative if and only if there exists an invertiblematrix P such that P-1SP(?)Tn(F).Theorem 3.8 Let F be an algebraically closed field. Then a subring S of M3(F) is weakly semicommutative, then either S is reduced or Sisomorphic to a subring of block triangular matrix ring.Direct product of a finite number of weakly semicommutative rings is weakly semicommutative. However, an infinite direct sum of weakly semicommutative rings is not necessarily weakly semicommutative.Example 3.10 Let F be a field, S=M2(F),Let R=multiply from n=1 to +∞Un.Then R is not weakly semicommutative. The polynomial rings over semicommutative rings are not necessarily semicommutative.Theorem 4.4 The polynomial rings over semicommutative rings are weakly semicommutative.
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