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Let $R$ be a ring (commutative, with unit). Show that $A=\{\sum a_iT^i\in R[T] \; : \; a_1=0\}$ is a subring of $R[T]$ and isomorphic to $R[X][Y]/(X^2-Y^3)$.

Of course, I'm trying to find a ring homomorphism of $R[X][Y]$ into $R[T]$ with image $A$ and kernel $(X^2-Y^3)$. Can you give me a hint?

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    Related: http://math.stackexchange.com/questions/11305342017-02-05

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How about $X \mapsto T^3$ and $Y \mapsto T^2$?

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    @Stefan, it's related to the [Frobenius problem](http://en.wikipedia.org/wiki/Frobenius_problem). The set $\mathbb{N}a+\mathbb{N}b$ contains all numbers greater than a certain limit and some less than this limit, but I'm not sure the smaller set is easy to describe.2011-05-19
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If you choose a basis for $R[X][Y]/(X^2-Y^3)$ (as a free $R$-module), you will have a clearer image of what it looks like. For example you can show that $(1,Y,Y^2,Y^3,\ldots,X,XY,XY^2,XY^3,\ldots)$ is one possible basis.

And then look at what is the image of the basis under the isomorphism.