$R$ module homomorphisms between two modules

Solution 1:

No and no, to the questions in the first line.

1. $k[x,y]/(x,y)\cong k$, so we can define a map by sending $(p,0)\mapsto p\cdot(ax+by)$ and $(0,q)\mapsto q\cdot(cx+dy)$ for any $a,b,c,d,p,q\in k$. Each of these maps are $R$-linear, and any such map is non-zero as long as $a,b,c,d$ are not all zero.

2. $(x,y)/(x,y)^2\cong k\langle x,y\rangle$, where the RHS is the $k$-vector space on the basis $\{x,y\}$. To see this, the polynomials in $(x,y)$ are exactly those with zero constant term, and any two such polynomials with the same linear term agree up to an element of $(x,y)^2=(x^2,xy,y^2)$ which consists of all polynomials with zero constant and linear terms. As $k[x,y]/(x,y)$ and $(x,y)/(x,y)^2$ have different dimensions as $k$-vector spaces, they cannot be isomorphic.

As for the third question at the end, the above should give you the tools to work on that too. You're asking about maps $k\to k^2$, and any such map is determined by the vector which is the image of $1\in k$.