Weak Mccoy Rings And Right Symmetric Rings

We have four parts in this paper:In the first part we introduce the grand results in the reversible ring, symmetric ring, semi-commutative ring, weak McCoy ring, skew polynomial ring and differential polynomial ring in this paper.In the second part we generalize the concept of McCoy ring, and pose the concept of weak McCoy ring. Meanwhile some properties and extensions are investigated. The following statements are the main results:Theorem2.2.2If R is semicommutative ring,then R is weak McCoy ring (?) R[x] is weak McCoy ring.Proposition2.2.3For a ring R,the following statements are equivalent:(1) R is weak McCoy ring;(2) eR and (1-e)R are weak McCoy and e is a central idempotent element which meet e≠0.In the third part we generalize the weak McCoy properties of skew polynomial rings and differential polynomial rings.The following statements are the results:Theorem3.1.7Let R is reversible ring, α∈end(R),and for arbitrary a,b∈R,ab=0(?) aα(b)=0,then R is weak McCoy ring (?) R[x;a] is weak McCoy ring.Theorem3.2.6Let R is reversible ring,δ is the derivation of R, and for arbitrary a,b∈R,ab=0(?) aδ(b)=0,then R is weak McCoy ring (?) R[x;δ] is weak McCoy ring.In the fourth part we generalize the concept of left Symmetric ring, and pose the concept of right Symmetric ring, some examples are provided. Also we investigate some properties and extensions. The following statements are the results:Theorem4.7Let R/I is right Symmetric ring, I△R,if I is reduced, R is right Symmetric ring.Theorem4.8R is right Symmetric ring,I1,I2, are the left and right annihilator of R, then R/I1, R/I2are right Symmetric ringsTheorem4.9R is Armendariz ring, the following statement are equivalent:(1) R is right Symmetric ring.(2) R[x] is right Symmetric ring.Theorem4.10If R is reduced ring,R[x]/(xn) iS right Symmetric ring, then (xn) is the ideal generated by xn.