Are Indecomposable objects preserved under exact full and faithful functors?

Question

Hello everyone I am new in the site, and I have the following question:

Let $A,B$ be abelian categories and let $F:A\rightarrow B$ be an exact full and faithful functor. Is it true that if $M$ is an indecomposable object in $A$, then $F(M)$ is an indecomposable object in $B$?

I know the answer is yes if $F$ indeed induces an equivalence of categories, and more generally, whenever $F(A)$ is a Serre subcategory of $B$, but I wonder if it is true in general and, in that case, how to prove it.

Thanks in advance.

Answer 1

Yes, and the assumption of exactness is unnecessary. Note that an object $M$ in an abelian category is decomposable iff there exist more than two $f:M\to M$ such that $f^2=f$ (such $f$s are exactly the projections onto direct summands of $M$, and a nontrivial direct summand is one with $f\neq 0,1$). Whether more than two such $f$ exist is obviously preserved by any fully faithful functor.