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We study the family of irreducible modules for quantum affine$\lie{sl}_{n+1}$ whose Drinfeld polynomials are supported on just one node ofthe Dynkin diagram. We identify all the prime modules in this family and provea unique factorization theorem. The Drinfeld polynomials of the prime modulesencode information coming from the points of reducibility of tensor products ofthe fundamental modules associated to $A_m$ with $m\le n$. These prime modulesare a special class of the snake modules studied by Mukhin and Young. We relateour modules to the work of Hernandez and Leclerc and define generalizations ofthe category $\mathscr C^-$. This leads naturally to the notion of an inflationof the corresponding Grothendieck ring. In the last section we show that thetensor product of a (higher order) Kirillov--Reshetikhin module with its dualalways contains an imaginary module in its Jordan--Holder series and give anexplicit formula for its Drinfeld polynomial. Together with the results of\cite{HL13a} this gives examples of a product of cluster variables which arenot in the span of cluster monomials. We also discuss the connection of ourwork with the examples arising from the work of \cite{LM18}.  Finally, we use our methods to give a family of imaginary modules in type$D_4$ which do not arise from an embedding of $A_r$ with $r\le 3$ in $D_4$.

Higher order Kirillov-Reshetikhin modules, Imaginary modules and Monoidal Categorification for $U_q(A_n^{(1)})$