Subrings and quotients of finite semigroup algebras

Question

Let $F = \Bbb F_p$ be a prime finite field and $R$ an arbitrary finite-dimensional associative (+ let's say unital) algebra over $F$. Then $R$ is a subalgebra (=subring here) of a matrix algebra $M_n(F)$ for some $n$, and $M_n(F)$ is a quotient of a certain semigroup algebra $FS$ with $S$ finite. (See e.g. this and this.)

My questions:

1. does there necessarily exist a finite semigroup $S$ such that $R$ is a subring of $FS$?
2. and 3): same, but with "subring" replaced with "quotient" and "direct factor" (as in $FS \cong R \times T$).

Point 2) would have been obvious were we to allow infinite semigroups, as the free algebra is a semigroup algebra.