Hom of algebras For any $R$-algebras $A$ and $B$, 
doea their set of R-algebra morphisms $\mathrm{Hom}_{R_{\mathrm{Alg}}}(A,B)$ necessarily again have the strucutre of an $R$-algebra?

The set $\mathrm{Hom}_{R-{\mathrm{Alg}}}(A,B)$ is never an algebra if $B\neq 0$ because the zero map is not a morphism.
Indeed, the zero map sends $1_A$ to $0_B$ instead of sending $1_A$ to $1_B$ like any honest morphism should.
