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So I think this question is trivial but I can't seem to be able to do it so here we go : what is the tensor product

$$k[x,y]/(y^2-x^3) \otimes_{k[y]} k[x,y]/(y^2-x^3)\ ?$$

My guess is that it is $k[x_1,x_2,y]/(y^2-x_1^3,y^2-x_2^3)$, but sadly it's just a guess.

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Tchichikov
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By using this result we get $$(R[X_1]/I)\otimes_R(R[X_2]/J)\simeq R[X_1,X_2]/(I+J)R[X_1,X_2].$$ Now set $R=k[Y]$, $I=(X_1^3-Y^2)$ and $J=(X_2^3-Y^2)$ and find your guess.

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