A note on the amenability constant of Banach algebras

Given a Banach algebra $A$, with an amenable ideal $I$ and amenable quotient $A/I$, we seek for relations among the amenability constants of $A$, $A/I$ and $I$. We also provide some examples as applications. In particular, we propose a suitable approach to the amenability constant of $A\#$. Finally, we give an upper bound for the amenability constant of the augmentation ideal $L^1_0(G)$ of an amenable $\sigma$-compact group $G$.