Is $C[x,y,z]/<y-x^2,z-x^3>$ an Integral Domain?

In summary, an integral domain is a commutative ring with no zero divisors, similar to a field but without multiplicative inverses for all elements. The quotient ring $C[x,y,z]/<y-x^2,z-x^3>$ is a structure created by factoring out certain polynomials in the polynomial ring $C[x,y,z]$, and it is not an integral domain due to the presence of zero divisors. Studying this quotient ring and other quotient rings has important implications in various fields, such as algebraic geometry and physics.
  • #1
evinda
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Hi! (Smile)

Can I justify like that, that $C[x,y,z]/<y-x^2,z-x^3>$ is an integral domain?

We show that $ker(\phi)=<y-x^2,z-x^3>$.

From the theorem of isomorphism, we have that $C[x,y,z]/<y-x^2,z-x^3> \cong im(\phi)$

$im(\phi)$ is a subring of $\mathbb{C}[x]$

$\mathbb{C}[x]$ is an integral domain, since $\mathbb{C}$ is a field.

As $im(\phi)$ is a subring of $\mathbb{C}[x]$, $im(\phi)$ is also an integral domain.

(Thinking)
 
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  • #2
Hi evinda,

You need to define some morphism to claim such things.
 

FAQ: Is $C[x,y,z]/<y-x^2,z-x^3>$ an Integral Domain?

What is an integral domain?

An integral domain is a mathematical structure in abstract algebra that is similar to a field, except that it does not necessarily have multiplicative inverses for all elements. It is a commutative ring with no zero divisors, meaning that if two elements multiply to equal zero, then at least one of the elements must be equal to zero.

How is an integral domain different from a field?

An integral domain is similar to a field, but it does not necessarily have multiplicative inverses for all elements. This means that not all elements have a unique multiplicative inverse, as they do in a field. Additionally, an integral domain does not necessarily have all the properties of a field, such as closure under division.

What is the quotient ring $C[x,y,z]/$?

The quotient ring $C[x,y,z]/$ is a mathematical structure created by taking the polynomial ring $C[x,y,z]$ and factoring out the ideal generated by the polynomials $y-x^2$ and $z-x^3$. This means that any polynomial in $C[x,y,z]$ that has both $y-x^2$ and $z-x^3$ as factors will be considered equivalent to the zero polynomial in the quotient ring.

Is $C[x,y,z]/$ an integral domain?

No, $C[x,y,z]/$ is not an integral domain. This is because the ideal $$ contains zero divisors, such as the elements $y-x^2$ and $z-x^3$. This means that not all elements in the quotient ring have unique multiplicative inverses, which is a property of integral domains.

What is the significance of studying the quotient ring $C[x,y,z]/$?

The quotient ring $C[x,y,z]/$ is a useful mathematical structure for studying certain geometric objects, such as algebraic varieties. It allows for the study of solutions to systems of polynomial equations, which has applications in fields such as physics, computer science, and engineering. Additionally, the study of quotient rings in general has important implications in abstract algebra and algebraic geometry.

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