• Communications of the Korean Mathematical Society
• /
• v.33 no.3
• /
• pp.721-739
• /
• 2018

In this paper, we investigate the problem of describing the form of Jordan generalized derivations on trivial extension algebras. One of the main results shows, under some conditions, that every Jordan generalized derivation on a trivial extension algebra is the sum of a generalized derivation and an antiderivation. This result extends the study of Jordan generalized derivations on triangular algebras (see [12]), and also it can be considered as a "generalized" counterpart of the results given on Jordan derivations of a trivial extension algebra (see [11]).

• Communications of the Korean Mathematical Society
• /
• v.31 no.2
• /
• pp.247-259
• /
• 2016

We first give the constructions of (Jordan) higher derivations on a trivial extension algebra and then we provide some sufficient conditions under which a Jordan higher derivation on a trivial extension algebra is a higher derivation. We then proceed to the trivial generalized matrix algebras as a special trivial extension algebra. As an application we characterize the construction of Jordan higher derivations on a triangular algebra. We also provide some illuminating examples of Jordan higher derivations on certain trivial extension algebras which are not higher derivations.

• Communications of the Korean Mathematical Society
• /
• v.21 no.1
• /
• pp.1-9
• /
• 2006

A ring R is called (left principally) quasi-Baer if the left annihilator of every (principal) left ideal of R is generated by an idempotent. Let R be a ring, G be an ordered monoid acting on R by $\beta$ and R be G-compatible. It is shown that R is (left principally) quasi-Baer if and only if skew monoid ring $R_{\beta}[G]$ is (left principally) quasi-Baer. If G is an abelian monoid, then R is (left principally) quasi-Baer if and only if the Cohn-Jordan extension $A(R,\;\beta)$ is (left principally) quasi-Baer if and only if left Ore quotient ring $G^{-1}R_{\beta}[G]$ is (left principally) quasi-Baer.

• Korean Journal of Mathematics
• /
• v.17 no.1
• /
• pp.37-41
• /
• 2009

We would like to propose Dirichlet-Jordan theorem on the space of summable in measure(SIM). Surely, this is a kind of extension of bounded variation([1, 4]), and considered as an application of fuzzy set such that ${\alpha}$-cut is 0.

• Journal of the Chungcheong Mathematical Society
• /
• v.23 no.4
• /
• pp.741-748
• /
• 2010

Let R be a 2-torsionfree prime ring. Our goal in this note is to show that the existence of a nonzero Jordan higher left derivation on R implies R is commutative. This result is used to prove a noncommutative extension of the classical Singer-Wermer theorem in the sense of higher derivations.

• Honam Mathematical Journal
• /
• v.43 no.1
• /
• pp.68-77
• /
• 2021

This paper considers pseudo Jordan homomorphisms on module extensions of Banach algebras and triangular Banach algebras. We characterize pseudo Jordan homomorphisms on module extensions of Banach algebras and triangular Banach algebras. Moreover, we define pseudo derivations on the above stated Banach algebras and characterize this new notion on those algebras.

• Communications of the Korean Mathematical Society
• /
• v.39 no.2
• /
• pp.373-397
• /
• 2024

Lambek introduced the concept of symmetric rings to expand the commutative ideal theory to noncommutative rings. In this study, we propose an extension of symmetric rings called strongly α-symmetric rings, which serves as both a generalization of strongly symmetric rings and an extension of symmetric rings. We define a ring R as strongly α-symmetric if the skew polynomial ring R[x; α] is symmetric. Consequently, we provide proofs for previously established outcomes regarding symmetric and strongly symmetric rings, directly derived from the results we have obtained. Furthermore, we explore various properties and extensions of strongly α-symmetric rings.

• 전기의세계
• /
• v.22 no.4
• /
• pp.25-29
• /
• 1973

The computation of transmittances between arbitrary input and output nodes is of particular interest in the signal flow graph theory imput. The signal flow matrix [T] can be defined by [X]=-[T][X] where [X] and [Y] are input nose and output node matrices, respectively. In this paper, the followings are discussed; 1) Reduction of nodes by reforming the signal flow matrix., 2) Solution of input-output relationships by means of Gauss-Jordan reduction method, 3) Extension of the above method to the matrix signal flow graph.

• Communications of the Korean Mathematical Society
• /
• v.33 no.4
• /
• pp.1123-1123
• /
• 2018

• 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.