Let $A,B,C$ be algebras over a field $F$ ($F=\mathbb{Q}$ or $\mathbb{Z}/p$, $p$ prime). The height of $A$ is defined to be $$ \mathrm{height}(A)=\sup_{a\in A}\inf\{n(a)\in \mathbb{N}\mid a^{n(a)+1}=0 \}. $$ Let $1$ be the multiplicative unit. Suppose there is an exact sequence of algebra homomorphisms preserving unit $$ 1\to A\to B\to C\to 1. $$ (all the algebras are cohomology rings $H^*(M;\mathbb{Z}_2)$ and homomorphisms are induced by continuous maps between manifolds. I find the short exact sequence on Mapping class groups and function spaces, C.F. Bodigheimer, F.R. Cohen, M.D. Peim, Contemporary Mathematics, Vol. 271, 2001, page 29, Corollary 7.4.)
Question 1: When the sequence splits, I want to express $B$ in terms of $A,C$. Can we obtain $$ B=A\oplus C? $$ Is there any difference if I change the $1$'s in the above exact sequence to $0$?
Question 2: Can we obtain $$ \mathrm{height}(B)=\max\{\mathrm{height}(A),\mathrm{height}(C)\} $$ or $$ \mathrm{height}(B)= \mathrm{height}(A)+\mathrm{height}(C)? $$ Thanks!