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Show that in the ring $\mathbb Q[x, y]/(x^3 -y^2)$ the element $x +(x^3 -y^2)$ is irreducible but not prime.

Not sure how to show this. I know that $(x^3 -y^2)$ is a prime ideal but I cannot continue. How should I proceed?

Rócherz
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bensimmonsisarookie
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    What are your thoughts on the problem? What have you tried and where did you get stuck? Also, what are you definitions of prime and irreducible? – Servaes Mar 05 '19 at 15:02
  • @Servaes Only in non-domains does there arise any variation in such definitions. – Bill Dubuque Mar 05 '19 at 15:23
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    @Crostul My point is that there are unique definitions of irreducible elements in domains, but the definitions bifurcate in non-domians. SInce the OP's ring is a domain, there is no need to ask which definition is being used. – Bill Dubuque Mar 05 '19 at 15:27
  • Use the definition of a prime element. You have to show that the element $\alpha = x + (x^3-y^2)$ is prime in the ring $R = \mathbb Q[x,y] / (x^3 - y^2)$. Now, an element is prime if the ideal generated by it is prime, which means that if you quotient it out you get an integral domain. So what you have to prove is that $R / \langle \alpha\rangle$ is an integral domain. Can you use some isomorphism theorem to clarify what this quotient is? For non-irreducibility, try to use the fact that you can "absorb" $x^3-y^2$ to find two non-unit elements whose product is $x$ modulo $x^3-y^2$. – Sarvesh Ravichandran Iyer Mar 05 '19 at 16:02
  • @BillDubuque There are different ways to phrase the same definition; for example an element $p\in R$ is prime if $p\mid xy\implies p\mid x\vee p\mid y$, or equivalently if $R/(p)$ is a domain. Considering the elementary level of the question, I didn't want to assume that OP is familiar with the fact that these two are equivalent. – Servaes Mar 05 '19 at 16:34
  • @Servaes Do you actually know any textbooks that use the 2nd as a definition of a prime element? – Bill Dubuque Mar 05 '19 at 16:58
  • @BillDubuque It's the definition I was taught in my first year algebra course (we didn't have a textbook, just lecture notes), with the immediate follow-up that this is equivalent to the first definition, hence the terminology. – Servaes Mar 05 '19 at 17:15

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