Infinite dimensional representation such that every subrepresentation is reducible Let $V$ be a nonzero finite dimensional representation of an algebra $A$. 
  
  a) Show that
  it has an irreducible subrepresentation. 
  
  b) Show by example that this does not always hold for infinite dimensional representations.


I did not have any problems with part a), but I'm struggling to find an example for part b). Any help?
Let $A=\mathbb{C}[x]$ and consider the regular representation of $A$ acting on itself. Every nonzero element generates a subrepresentation isomorphic to $A$, so this definitely does not have any irreducible subrepresentations.
I'm not quite sure if this contraption is correct, but I'll give it a go, because I'm also interested in this question.

b) Consider an algebra $A = \mathbb{Z}$ and its infinite-dimensional representation $V\subset\mathbb{Z^\infty}$ defined by recurrence relation $c_{i+1} = c_{i} + 1$ together with a homomorphism of algebras $\rho: A \rightarrow EndV$ defined by an operator $T$ such that:

$$\rho(a) = T_a: (c_i,c_{i+1},c_{i+2},\dots) \rightarrow (c_{i+a},c_{i+a+1},c_{i+a+2},\dots)$$

Irreducibility of this representation follows immediately from the observation that every $\rho(a)$, $a \in A$ has an infinite image.