  We show that the multiples of the backward shift operator on the spaces
$\ell_{p}$, $1\leq p<\infty$, or $c_{0}$, when endowed with coordinatewise
multiplication, do not possess frequently hypercyclic algebras. More generally,
we characterize the existence of algebras of $\mathcal{A}$-hypercyclic vectors
for these operators. We also show that the differentiation operator on the
space of entire functions, when endowed with the Hadamard product, does not
possess frequently hypercyclic algebras. On the other hand, we show that for
any frequently hypercyclic operator $T$ on any Banach space, $FHC(T)$ is
algebrable for a suitable product, and in some cases it is even strongly
algebrable.
