• Bulletin of the Korean Mathematical Society
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• v.20 no.1
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• pp.31-36
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• 1983

Let R be a commutative ring with 1, and A a unitary commutative R-algebra. By a derivation module of A, we mean a pair (M, d), where M is an A-module and d: A.rarw.M and R-derivation, i.e., d is an R-linear mapping such that d(ab)=a)db)+b(da). A derivation module homomorphism f:(M,d).rarw.(N, .delta.) is an A-homomorphism f:M.rarw.N such that f.d=.delta.. A derivation module of A, (U, d), there exists a unique derivation module homomorphism f:(U, d).rarw.(M,.delta.). In fact, a universal derivation module of A exists in the category of derivation modules of A, and is unique up to unique derivation module isomorphisms [2, pp. 101]. When (U,d) is a universal derivation module of R-algebra A, the A-module U is denoted by U(A/R). For out convenience, U(A/R) will also be called a universal derivation module of A, and d the R-derivation corresponding to U(A/R).

• The Pure and Applied Mathematics
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• v.23 no.4
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• pp.385-387
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• 2016

$Fo{\check{s}}ner$ [4] defined a generalized module left (m, n)-derivation and proved the Hyers-Ulam stability of generalized module left (m, n)-derivations. In this note, we prove that every generalized module left (m, n)-derivation is trival if the algebra is unital and $m{\neq}n$.

• The Pure and Applied Mathematics
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• v.24 no.1
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• pp.33-34
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• 2017

$Fo{\check{s}}ner$ [1] defined a module left (m, n)-derivation and proved the Hyers-Ulam stability of module left (m, n)-derivations. In this note, we prove that every module left (m, n)-derivation is trival if the algebra is unital and $m{\neq}n$.

• The Pure and Applied Mathematics
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• v.16 no.1
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• pp.99-106
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• 2009

In this paper, we improve the generalized Hyers-Ulam stability and the superstability of module left derivations due to the results of [7].

• Bulletin of the Korean Mathematical Society
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• v.36 no.2
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• pp.247-257
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• 1999

In this article, we give an example of local derivation, that is not derivation, on the algebra F(x1,…, xn) of rational functions in x1, …, xn over an infinite field F, and show that if X is a set of symbols and {x1,…, xn} is a finite subset of X, n$\geq$1, then each local derivation of F[x1,…, xn] into F[X] is a F-derivation and each local derivation of F[X] into itself is also a F-derivation.

• Communications of the Korean Mathematical Society
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• v.35 no.3
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• pp.891-906
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• 2020

In this paper, the commutative module amenable Banach algebras are characterized. The hereditary and permanence properties of module amenability and the relations between module amenability of a Banach algebra and its ideals are explored. Analogous to the classical case of amenability, it is shown that the projective tensor product and direct sum of module amenable Banach algebras are again module amenable. By an application of Ryll-Nardzewski fixed point theorem, it is shown that for an inverse semigroup S, every module derivation of 𝑙¹(S) into a reflexive module is inner.

• Journal of Intelligence and Information Systems
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• v.7 no.2
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• pp.65-81
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• 2001

This paper presets the process of knowledge aquisition by partitioning the phase of analysis & design for knowledge-base module construction of Expert System into five steps to derive rule systematically. Also, this paper presents the process and the task that knowledge engineer must do work each step. The knowledge-base module of STAFS expert system was constructed by considering the destruction rate and the commander\`s intention using the proposed rule derivation methodology.

• Bulletin of the Korean Mathematical Society
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• v.33 no.2
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• pp.221-228
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• 1996

Throughout, R will represent an associative ring with center Z(R). A module X is said to be n-torsionfree, where n is an integer, if nx = 0, $x \in X$ implies x = 0. An additive mapping $D : R \to X$, where X is a left R-module, will be called a Jordan left derivation if $D(a^2) = 2aD(a), a \in R$. M. Bresar and J. Vukman [1] showed that the existence of a nonzero Jordan left derivation of R into X implies R is commutative if X is a 2-torsionfree and 3-torsionfree left R-module.

• Journal for History of Mathematics
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• v.17 no.2
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• pp.97-103
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• 2004

In this paper, we will prove that a free join algebra and a universal derivation module of its subalgebras have a universal derivation module induced by its subalgebras.

• Journal of the Chungcheong Mathematical Society
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• v.6 no.1
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• pp.65-69
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• 1993

In this paper, we show that the module derivation D is continuous on the Banach algebra and the Silov algebra, and also that the derivation restricted by separating space and the radical on the semi prime Banach algebra is continuous.