= Solution
Coordinatewise multiplication gives $V=A(1,1,0)$, so \b[the third <module> is cyclic].
A <submodule> of a <cyclic module> need not be cyclic. In the second example the first-row space is a <left ideal> and hence a <submodule> of the regular <module> $A$. The regular <module> is cyclic, generated by $1_A$, whereas its first-row <submodule> has just been shown not to be cyclic. Thus \b[the proposed inheritance property fails even for a finite-dimensional <algebra>].
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