  Every ring extension of $A$ by $R$ induces a pair of group homomorphisms
$\mathcal{L}^{*}:R\to End_\Z(A)/L(A);\mathcal{R}^{*}:R\to End_\Z(A)/R(A),$
preserving multiplication, satisfying some certain conditions. A such 4-tuple
$(R,A,\mathcal{L}^{*},\mathcal{R}^{*})$ is called a ring pre-extension. Each
ring pre-extension induces a $R$-bimodule structure on bicenter $K_A$ of ring
$A,$ and induces an obstruction $k,$ which is a 3-cocycle of $\Z$-algebra $R,$
with coefficients in $R$-bimodule $K_A$ in the sense of Shukla. Each
obstruction $k$ in this sense induces a structure of a regular Ann-category of
type $(R,K_A).$ This result gives us the first application of Ann-category in
extension problems of algebraic structures, as well as in cohomology theories.
