Coproduct of Crossed A-Modules of R-Algebroids

Abstract

Abstract:In this study we construct, in the category XAlg(R) /A of crossed A-modules of R-algebroids, the coproduct of given two crossed A-modules M = (µ : M −→ A) and N = (η : N −→ A) of R-algebroids in two di erent ways: Firstly we construct the coproduct M • * N by using the free product M * N of pre-R-algebroids M and N, and then we construct the coproduct M • N by using the semidirect product M N of M and N via µ. Finally we construct an isomorphism between M • * N and M • N.

On generators of Peiffer ideal of a pre-$-algebroid in a precrossed module and applications

“…In [6], we've constructed in XAlg (R) /A the coproduct of given two crossed A-modules M = (µ : M −→ A) and N = (η : N −→ A) of R-algebroids by two different methods.…”

“…In [4], to construct a crossed module from a given precrossed module M = (µ : M −→ A) of R-algebroids, we've introduced the Peiffer ideal M, M of the pre-R-algebroid M where the procedure has given the functor (−)…”

“…In [6], on the other hand, we've constructed the coproduct of given two crossed A-modules M = (µ : M −→ A) and N = (η : N −→ A) of R-algebroids in two ways one of which depends basicly on dividing the free product M * N of pre-R-algebroids M and N with the ideal I M * N and the other on dividing their semidirect product M ⋉ N with the ideal I M⋉N .…”

“…In this study, we analyse the generators of M, M to determine their relations with the generators of M for further using and apply the outcomes to find the generators of the Peiffer ideals of pre-R-algebroids M * N and M ⋉ N, in [6], to reach the unsurprising result that I M * N and I M⋉N are in fact their Peiffer ideals, respectively. Throughout the paper R is a commutative ring.…”