I was reading the book on Commutative Algebra by Miles Reid and during the discussion of the zero divisors the authors make the following comment.
The ring $K[X,Y]/(XY)$ is a subring of $k[X] \oplus k[Y]$ with X and Y mapping to a non zero divisor and zero so that the product is zero.
The question is if the ring is a subring then which extra elements are there in $k[X] \oplus k[Y]$. What is the mapping ?
I think your direct sum $\oplus$ denotes the product $\times$ of rings. Furthermore any ring is a subring of itself.
In this case there is a well defined morphism of rings $k[X,Y]\rightarrow k[X]\times k[Y]$ induced by the assignments $X\mapsto (X,0)$ and $Y\mapsto (0,Y)$. It is clearly surjective and its kernel is given by the ideal $(XY)$. By the first isomorphism theorem $k[X,Y]/(XY) \cong k[X] \times k[Y]$.
