  We give a simple construction of the correspondence between square-zero
extensions $R'$ of a ring $R$ by an $R$-bimodule $M$ and second MacLane
cohomology classes of $R$ with coefficients in $M$ (the simplest non-trivial
case of the construction is $R=M=Z/p$, $R'=Z/p^2$, thus the Bokstein
homomorphism of the title). Following Jibladze and Pirashvili, we treat MacLane
cohomology as cohomology of non-additive endofunctors of the category of
projective $R$-modules. We explain how to describe liftings of $R$-modules and
complexes of $R$-modules to $R'$ in terms of data purely over $R$. We show that
if $R$ is commutative, then commutative square-zero extensions $R'$ correspond
to multiplicative extensions of endofunctors. We then explore in detail one
particular multiplicative non-additive endofunctor constructed from cyclic
powers of a module $V$ over a commutative ring $R$ annihilated by a prime $p$.
In this case, $R'$ is the second Witt vectors ring $W_2(R)$ considered as a
square-zero extension of $R$ by the Frobenius twist $R^{(1)}$.
