• Communications of the Korean Mathematical Society
• /
• v.25 no.3
• /
• pp.349-364
• /
• 2010

P. M. Cohn called a ring R reversible if whenever ab = 0, then ba = 0 for a, $b\;{\in}\;R$. Commutative rings and reduced rings are reversible. In this paper, we extend the reversible condition of a ring as follows: Let R be a ring and $\alpha$ an endomorphism of R, we say that R is right (resp., left) $\alpha$-shifting if whenever $a{\alpha}(b)\;=\;0$ (resp., $\alpha{a)b\;=\;0$) for a, $b\;{\in}\;R$, $b{\alpha}{a)\;=\;0$ (resp., $\alpha(b)a\;=\;0$); and the ring R is called $\alpha$-shifting if it is both left and right $\alpha$-shifting. We investigate characterizations of $\alpha$-shifting rings and their related properties, including the trivial extension, Jordan extension and Dorroh extension. In particular, it is shown that for an automorphism $\alpha$ of a ring R, R is right (resp., left) $\alpha$-shifting if and only if Q(R) is right (resp., left) $\bar{\alpha}$-shifting, whenever there exists the classical right quotient ring Q(R) of R.

• East Asian mathematical journal
• /
• v.14 no.2
• /
• pp.259-268
• /
• 1998

In this paper, for a commutative ring R with an identity, considering the endomorphism ring $End_R$(M) of left R-module $_RM$ which is (quasi-)injective or (quasi-)projective, some discriminations of prime endomorphism were found as follows: each epimorphism with the irreducible(or simple) kernel on a (quasi-)injective module and each monomorphism with maximal image on a (quasi-)projective module are prime. It was shown that for a field F, any given square matrix in $Mat_{n{\times}n}$(F) with maximal image and irreducible kernel is a prime matrix, furthermore, any given matrix in $Mat_{n{\times}n}$(F) for any field F can be factored into a product of prime matrices.

• Korean Journal of Mathematics
• /
• v.22 no.1
• /
• pp.151-167
• /
• 2014

Let M, N be modules over a ring R and $[M,N]=Hom_R(M,N)$. The concern is study of: (1) Some fundamental properties of [M, N] when [M, N] is regular or semipotent. (2) The substructures of [M, N] such as radical, the singular and co-singular ideals, the total and others has raised new questions for research in this area. New results obtained include necessary and sufficient conditions for [M, N] to be regular or semipotent. New substructures of [M, N] are studied and its relationship with the Tot of [M, N]. In this paper we show that, the endomorphism ring of a module M is regular if and only if the module M is semi-injective (projective) and the kernel (image) of every endomorphism is a direct summand.

• East Asian mathematical journal
• /
• v.18 no.2
• /
• pp.195-203
• /
• 2002

Let R be a prime ring with characteristic not two and U is a nonzero left ideal of R which contains no nonzero nilpotent right ideal as a ring. For a $(\sigma,\tau)$-derivation d : R$\rightarrow$R, we prove the following results: (1) If d is an endomorphism on R then d=0. (2) If d is an anti-endomorphism on R then d=0. (3) If d(xy)=d(yx), for all x, y$\in$R then R is commutative. (4) If d is an homomorphism or anti-homomorphism on U then d=0.

• Korean Journal of Mathematics
• /
• v.22 no.2
• /
• pp.217-234
• /
• 2014

Mason extended the reflexive property for subgroups to right ideals, and examined various connections between these and related concepts. A ring was usually called reflexive if the zero ideal satisfies the reflexive property. We here study this property skewed by ring endomorphisms, introducing the concept of an ${\alpha}$-skew reflexive ring, where is an endomorphism of a given ring.

• Bulletin of the Korean Mathematical Society
• /
• v.44 no.4
• /
• pp.777-788
• /
• 2007

An endomorphism ${\alpha}$ of a ring R is called right(left) symmetric if whenever abc=0 for a, b, c ${\in}$ R, $ac{\alpha}(b)=0({\alpha}(b)ac=0)$. A ring R is called right(left) ${\alpha}-symmetric$ if there exists a right(left) symmetric endomorphism ${\alpha}$ of R. The notion of an ${\alpha}-symmetric$ ring is a generalization of ${\alpha}-rigid$ rings as well as an extension of symmetric rings. We study characterizations of ${\alpha}-symmetric$ rings and their related properties including extensions. The relationship between ${\alpha}-symmetric$ rings and(extended) Armendariz rings is also investigated, consequently several known results relating to ${\alpha}-rigid$ and symmetric rings can be obtained as corollaries of our results.

• Communications of the Korean Mathematical Society
• /
• v.15 no.3
• /
• pp.453-467
• /
• 2000

For any S-flat module RM(which will be called endoflat) with a commutaitve ring R with identity, where S is the endomorphism ring RM, the fact that every epimorphism is an automorphism has been proved and the Jacobson Radical Rad(S) of S is described as follow; Rad(S) = { f$\in$S｜Imf=Mf is small in M} = {f$\in$S｜Imf $\leq$Rad(M)}. Additionally for any quasi-injective endo-flat module RM, the fact that every monomorphism is an automorphism has been proved and the Jacobson Radical Rad(S) for any quasi-injective endo-flat module has been studied too. Also some equivalent conditions for the semi-primitivity of any faithful endo-flat module RM with the open Jacobson Radical Rad(M) and those for the semi-simplicity of any faithful endo-flat quasi-injective module RM with the closed Socle Soc(M) have been studied.

• East Asian mathematical journal
• /
• v.10 no.2
• /
• pp.381-384
• /
• 1994

• East Asian mathematical journal
• /
• v.12 no.2
• /
• pp.213-233
• /
• 1996

• East Asian mathematical journal
• /
• v.5 no.1
• /
• pp.81-85
• /
• 1989