Change of Base for Commutative Algebras

In this paper we examine on a pair of adjoint functors $(\phi ^{\ast},\phi_{\ast})$ for a subcategory of the category of crossed modules over commutative algebras where ${\varphi }^{\ast }:\mathbf{X}\mathbf{M}\mathbf{o}\mathbf{d}$$\mathbf{\text{/}}$$% Q\rightarrow $ $\mathbf{X}\mathbf{M}\mathbf{o}\mathbf{d}\mathbf{/}P$, pullback, which enables us to move from crossed $Q$-modules to crossed $P$-modules by an algebra morphism $\phi :P\rightarrow Q$and$\phi_{\ast}:$\mathbf{X}\mathbf{M}\mathbf{o}\mathbf{d}$$\mathbf{\text{/}}$P\rightarrow $ $\mathbf{X}\mathbf{M}\mathbf{o}\mathbf{d}\mathbf{/}Q$, induced. We note that this adjoint functor pair $(\phi ^{\ast},\phi_{\ast})$makes$p:$\mathbf{X}\mathbf{M}\mathbf{o}\mathbf{d}$\rightarrow $ $k$$\mathbf{\text{-Alg}}$into a bifibred category over $k$$\mathbf{\text{-Alg}}$, the category of commutative algebras, where $p$ is given by $p(C,R,\mathrm{\partial })=R\mathrm{.}$ Also, some examples and results on induced crossed modules are given.