|
Structure of Abelian rings
Juncheol HAN,Yang LEE,Sangwon PARK
Front. Math. China. 2017, 12 (1): 117-134.
https://doi.org/10.1007/s11464-016-0586-z
Let R be a ring with identity. We use J(R), G(R), and X(R) to denote the Jacobson radical, the group of all units, and the set of all nonzero nonunits in R, respectively. A ring is said to be Abelian if every idempotent is central. It is shown, for an Abelian ring R and an idempotent-lifting ideal N ⊆ J(R) of R, that R has a complete set of primitive idempotents if and only if R/N has a complete set of primitive idempotents. The structure of an Abelian ring R is completely determined in relation with the local property when X(R) is a union of 2, 3, 4, and 5 orbits under the left regular action on X(R) by G(R). For a semiperfect ring R which is not local, it is shown that if G(R) is a cyclic group with 2 ∈ G(R), then R is finite. We lastly consider two sorts of conditions for G(R) to be an Abelian group.
References |
Related Articles |
Metrics
|
|
Quaternion rings and octonion rings
Gangyong LEE,Kiyoichi OSHIRO
Front. Math. China. 2017, 12 (1): 143-155.
https://doi.org/10.1007/s11464-016-0571-6
In this paper, for rings R, we introduce complex rings ℂ(R), quaternion rings ℍ(R), and octonion rings О, which are extension rings of R; R ⊂ ℂ(R) ⊂ ℍ(R) ⊂ O(R). Our main purpose of this paper is to show that if R is a Frobenius algebra, then these extension rings are Frobenius algebras and if R is a quasi-Frobenius ring, then ℂ(R) and ℍ(R) are quasi-Frobenius rings and, when Char(R) = 2, O(R) is also a quasi-Frobenius ring.
References |
Related Articles |
Metrics
|
|
Center construction and duality of category of Hom-Yetter-Drinfeld modules over monoidal Hom-Hopf algebras
Bingliang SHEN,Ling LIU
Front. Math. China. 2017, 12 (1): 177-197.
https://doi.org/10.1007/s11464-016-0594-z
*Abstract:Let (H,α) be a monoidal Hom-Hopf algebra. In this paper, we will study the category of Hom-Yetter-Drinfeld modules. First, we show that the category of left-left Hom-Yetter-Drinfeld modules HH H? Y? D is isomorphic to the center of the category of left (H,α)-Hom-modules. Also, by the center construction, we get that the categories of left-left, left-right, right-left, and right-right Hom-Yetter-Drinfeld modules are isomorphic as braided monoidal categories. Second, we prove that the category of finitely generated projective left-left Hom-Yetter-Drinfeld modules has left and right duality.
References |
Related Articles |
Metrics
|
|
New characterizations for core inverses in rings with involution
Sanzhang XU,Jianlong CHEN,Xiaoxiang ZHANG
Front. Math. China. 2017, 12 (1): 231-246.
https://doi.org/10.1007/s11464-016-0591-2
The core inverse for a complex matrix was introduced by O. M. Baksalary and G. Trenkler. D. S. Rakíc, N.Č. Diňcíc and D. S. Djordjevíc generalized the core inverse of a complex matrix to the case of an element in a ring. They also proved that the core inverse of an element in a ring can be characterized by five equations and every core invertible element is group invertible. It is natural to ask when a group invertible element is core invertible. In this paper, we will answer this question. Let R be a ring with involution, we will use three equations to characterize the core inverse of an element. That is, let a, b ∈ R. Then a ∈ R# with a# = b if and only if (ab)∗ = ab, ba2= a, and ab2= b. Finally, we investigate the additive property of two core invertible elements. Moreover, the formulae of the sum of two core invertible elements are presented.
References |
Related Articles |
Metrics
|
|
A note on generalized Lie derivations of prime rings
Nihan Baydar YARBIL,Nurcan ARGAC
Front. Math. China. 2017, 12 (1): 247-260.
https://doi.org/10.1007/s11464-016-0589-9
Let R be a prime ring of characteristic not 2, A be an additive subgroup of R, and F, T, D,K: A → R be additive maps such that F[x, y]) = F(x)y − yK(x) − T(y)x + xD(y) for all x, y ∈ A. Our aim is to deal with this functional identity when A is R itself or a noncentral Lie ideal of R. Eventually, we are able to describe the forms of the mappings F, T, D, and K in case A = R with deg(R)>3 and also in the case A is a noncentral Lie ideal and deg(R)>9. These enable us in return to characterize the forms of both generalized Lie derivations, D-Lie derivations and Lie centralizers of R under some mild assumptions. Finally, we give a generalization of Lie homomorphisms on Lie ideals.
References |
Related Articles |
Metrics
|
17 articles
|