Second cohomology of multiplicative Lie rings and Schreier's extension theory

Our goal of this paper is to develop an analogue of the theory of group extensions for multiplicative Lie rings. We first define a factor system of pair of multiplicative Lie rings $(L,A)$, which use to construct an extension of $A$ by $L$. Then we state Schreier's theorem for multiplicative Lie rings. We also use this notation to introduce second cohomology group of multiplicative Lie rings. Finally, we show that the equivalence classes of multiplicative Lie ring extensions can be identified with second cohomology group $ H^2(L,A, \rho, h)$, where $L$ acts on $A$ by $\rho$ and $h$.