A Characteristic of Co-derivation of -R--Comodule -M- Nikken Prima Puspita (a*) Indah Emilia Wijayanti(b)
a) Mathematics Department, Faculty of Mathematics and Sciences, Universitas Diponegoro, Searang, Indonesia, 50275
*nikkenprima[at]lecturer.undip.ac.id
b) Mathematics Department, Faculty of Mathematics and Natural Sciences, Universitas Gadjah Mada, Yogyakarta, Indonesia
Abstract
Let -R- be a ring and -M- a right and left module over -R-. We already know the derivation of ring and module categories. A linear map -\delta:R\rightarrow R- is a derivation of ring -R- if -\delta(a\cdot b)=a\delta(b)+\delta(a)b-, for any -a,b\in R-. Moreover, we can bring it to the modules category by considering the element of ring -R- as elements of -R--modules -R-. In GH. Abbaspour et al. (2005), if -\delta- is a derivation of -R- and -f:M\rightarrow N- is an -R--linear map of left -R--module, then an additive map -d:M\rightarrow N- is a -(\delta,f)--derivation if -d(rm)=rd(m)+\delta(r)f(m)-, for any -r\in R- and -m\in M-. The dualization of derivation on module theory is called co-derivation. Consider -R- is an -R--coalgebra and -M- is an -R--bicomodule. We will show that if -R- is a commutative ring, it implies -d: M\rightarrow R- is a co-derivation of -R--bicomodule -M- if and only if -d- is a zero map. Consequently, when -M=R-, -R- is a trivial clean coalgebra over itself.
