present a new generalized version of Cline's formula and Jacobson's lemma for g-Drazin inverses in a ring.These generalized results extend many known results,e.g.,Chen and Abdolyousefi[Generalized Jacobson'...present a new generalized version of Cline's formula and Jacobson's lemma for g-Drazin inverses in a ring.These generalized results extend many known results,e.g.,Chen and Abdolyousefi[Generalized Jacobson's lemma in a Banach algebra,Comm.Algebra 49(2021)3263-3272],and Yan and Zeng[The generalized inverses of the products of two elements in a ring,Turkish J.Math.44(2020)1744-1756].展开更多
A ring R is a QB-ring provided that aR ＋ bR = R with a, b E R implies that there exists a y E R such that a＋ by ∈ Rq^-1. It is said that a ring R is a JB-ring provided that R/J（R） is a QB-ring, where J（R） is th...A ring R is a QB-ring provided that aR ＋ bR = R with a, b E R implies that there exists a y E R such that a＋ by ∈ Rq^-1. It is said that a ring R is a JB-ring provided that R/J（R） is a QB-ring, where J（R） is the Jacobson radical of R. In this paper, various necessary and sufficient conditions, under which a ring is a JB-ring, are established. It is proved that JB-rings can be characterized by pseudo-similarity. Furthermore, the author proves that R is a JB-ring iff so is R/J（R）^2.展开更多
A ring R is Zhou nil-clean if every element in R is the sum of two tripotents and a nilpotent that commute. Homomorphic images of Zhou nil-clean rings are explored. We prove that a ring R is Zhou nil-clean if and only...A ring R is Zhou nil-clean if every element in R is the sum of two tripotents and a nilpotent that commute. Homomorphic images of Zhou nil-clean rings are explored. We prove that a ring R is Zhou nil-clean if and only if 30 ∈ R is nilpotent and R/30R is Zhou nil-clean, if and only if R/BM（R） is 5-potent and BM（R） is nil, if and only if J（R） is nil and R/J（R） is isomorphic to a Boolean ring, a Yaqub ring, a Bell ring or a direct product of such rings. By means of homomorphic images, we completely determine when the generalized matrix ring is Zhou nil-clean. We prove that the generalized matrix ring Mn（R; s） is Zhou nil-clean if and only if R is Zhou nil-clean and s ∈ J（R）.展开更多
The necessary and sufficient conditions under which a ring satisfies regular power-substitution are investigated. It is shown that a ring R satisfies regular powersubstitution if and only if a-b in R implies that ther...The necessary and sufficient conditions under which a ring satisfies regular power-substitution are investigated. It is shown that a ring R satisfies regular powersubstitution if and only if a-b in R implies that there exist n ∈ N and a U E GLn （R） such that aU = Ub if and only if for any regular x ∈ R there exist m,n ∈ N and U ∈ GLn（R） such that x^mIn = xmUx^m, where a-b means that there exists x,y, z∈ R such that a =ybx, b = xaz and x= xyx = xzx. It is proved that every directly finite simple ring satisfies regular power-substitution. Some applications for stably free R-modules are also obtained.展开更多
An ideal I of a ring R is strongly separative provided that for all finitely generated projective R-modules A, B with A = AI and B = BI, if 2A ≌ A ＋ B, then A ≌ B. We prove in this paper that a regular ideal I of a...An ideal I of a ring R is strongly separative provided that for all finitely generated projective R-modules A, B with A = AI and B = BI, if 2A ≌ A ＋ B, then A ≌ B. We prove in this paper that a regular ideal I of a ring R is strongly separative if and only if each a E 1 ＋ I satisfying （1 - α）R ∝ r（a） is unit-regular, if and only if each a ∈ 1 ＋ I satisfying （1 - a2）R ∝ r（a2） is unit-regular, if and only if each a E 1 4- I satisfying R（1 - a）R = Rr（a） is unit-regular, if and only if each a ∈ 1 ＋ I satisfying R（1 -a^2）R = Rr（a^2） is unit-regular.展开更多
We get criteria of strong cleanness for several classes of 2 × 2 matrices over integers. For commutative local domains, we establish ones in terms of solvability of quadratic equations. Strongly clean matrices ov...We get criteria of strong cleanness for several classes of 2 × 2 matrices over integers. For commutative local domains, we establish ones in terms of solvability of quadratic equations. Strongly clean matrices over power series are also studied.展开更多
基金supported by the Natural Science Foundation of Zhejiang Province,China(No.LY21A010018).
文摘present a new generalized version of Cline's formula and Jacobson's lemma for g-Drazin inverses in a ring.These generalized results extend many known results,e.g.,Chen and Abdolyousefi[Generalized Jacobson's lemma in a Banach algebra,Comm.Algebra 49(2021)3263-3272],and Yan and Zeng[The generalized inverses of the products of two elements in a ring,Turkish J.Math.44(2020)1744-1756].
文摘A ring R is a QB-ring provided that aR ＋ bR = R with a, b E R implies that there exists a y E R such that a＋ by ∈ Rq^-1. It is said that a ring R is a JB-ring provided that R/J（R） is a QB-ring, where J（R） is the Jacobson radical of R. In this paper, various necessary and sufficient conditions, under which a ring is a JB-ring, are established. It is proved that JB-rings can be characterized by pseudo-similarity. Furthermore, the author proves that R is a JB-ring iff so is R/J（R）^2.
基金The authors are grateful to the referee for his/her careful the paper, and for the invaluable comments which improve our presentation reading of author H.Y. Chen was supported by the Natural Science Foundation of Zhejiang （No. LY17A010018）, China. The first Province
文摘A ring R is Zhou nil-clean if every element in R is the sum of two tripotents and a nilpotent that commute. Homomorphic images of Zhou nil-clean rings are explored. We prove that a ring R is Zhou nil-clean if and only if 30 ∈ R is nilpotent and R/30R is Zhou nil-clean, if and only if R/BM（R） is 5-potent and BM（R） is nil, if and only if J（R） is nil and R/J（R） is isomorphic to a Boolean ring, a Yaqub ring, a Bell ring or a direct product of such rings. By means of homomorphic images, we completely determine when the generalized matrix ring is Zhou nil-clean. We prove that the generalized matrix ring Mn（R; s） is Zhou nil-clean if and only if R is Zhou nil-clean and s ∈ J（R）.
基金supported by the grant of Hangzhou Normal University (No.200901).
文摘The necessary and sufficient conditions under which a ring satisfies regular power-substitution are investigated. It is shown that a ring R satisfies regular powersubstitution if and only if a-b in R implies that there exist n ∈ N and a U E GLn （R） such that aU = Ub if and only if for any regular x ∈ R there exist m,n ∈ N and U ∈ GLn（R） such that x^mIn = xmUx^m, where a-b means that there exists x,y, z∈ R such that a =ybx, b = xaz and x= xyx = xzx. It is proved that every directly finite simple ring satisfies regular power-substitution. Some applications for stably free R-modules are also obtained.
基金This research was supported by the Natural Science Foundation of Zhejiang Province （LY13A010019）, China.
文摘An ideal I of a ring R is strongly separative provided that for all finitely generated projective R-modules A, B with A = AI and B = BI, if 2A ≌ A ＋ B, then A ≌ B. We prove in this paper that a regular ideal I of a ring R is strongly separative if and only if each a E 1 ＋ I satisfying （1 - α）R ∝ r（a） is unit-regular, if and only if each a ∈ 1 ＋ I satisfying （1 - a2）R ∝ r（a2） is unit-regular, if and only if each a E 1 4- I satisfying R（1 - a）R = Rr（a） is unit-regular, if and only if each a ∈ 1 ＋ I satisfying R（1 -a^2）R = Rr（a^2） is unit-regular.
基金The research of the author was supported by the Natural Science Foundation of Zhejiang Province （LY13A010019） and the Fund of Hangzhou Normal University, China.
文摘We get criteria of strong cleanness for several classes of 2 × 2 matrices over integers. For commutative local domains, we establish ones in terms of solvability of quadratic equations. Strongly clean matrices over power series are also studied.
