Nil-cleanness Of 2×2 Formal Matrix Rings Over Commutative Rings

In matrix ring theory,professor Zhou Yiqiang and professor Tang Gaohua published a paper,named A class of formal matrix rings,which mainly introduced the formal matrix ring Mn(R;S)on R defined by the central elements s and its properties.Inspired by this idea,the nil-cleanness and strongly nil-cleanness in formal matrix ring over commutative rings are discussed in this paper,and get some conclusionsFirstly,the paper gives the equivalent characterization that the formal matrix ring over commutative rings is nil-clean ring,and proves that when F is a domain,Mn(F;s)is nil-clean ring if and only if F is a binary field.At the same time,it gives the specific form of nil-clean element in the 2×2 formal matrix ring over integral rings,and introduces the concept of strongly nil-clean ringThen,we give the equivalent characterization of the 2x2 formal matrix ring is strongly nil-clean ring over commutative local rings and define the upper triangular formal matrix ring.We discuss the specific form of its strongly nil-clean elements in the 2×2 and 3×3 upper triangular formal matrix ringFinally,we define a kind of involution operation*in the ring so as to give the concept nil*-clean ring on the ring with involution operation Then we continue to discuss the nil*-cleanness and strongly nil*-cleanness over commutative rings,that is,we find the conditions of their equivalence characterization and the specific example of nil*-clean ring and strongly nil*-clean ring.