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
Gold Member
MHB
3,836
0
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)
 
Physics news on Phys.org
  • #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.

Similar threads

I Proof for subgroup -- How prove it is a subgroup of Z^m?
Replies
11
Views
2K
MHB Show that the subspace U is ϕ^z-invariant
Replies
4
Views
1K
MHB Proving Equivalence of Statements Involving Endomorphisms - Can We Take $v=x$?
Replies
10
Views
3K
MHB Is the Matrix A Diagonalizable with Real Eigenvalues?
Replies
10
Views
1K
MHB Integral Domains and Characteristics - Char(D)
Replies
10
Views
3K
MHB Linear transformation and its matrix
Replies
8
Views
3K
MHB Subsets of $\mathbb{R}^2$ Satisfying S2 and S3 but Not S1: Empty Set
Replies
6
Views
2K
MHB (x) and (x,y) are prime ideals of Q[x,y]
Replies
4
Views
4K
I Finite fields, irreducible polynomial and minimal polynomial theorem
Replies
6
Views
2K
MHB The axioms of a vector space are satisfied
Replies
10
Views
2K

Hot Threads

Recent Insights

Back
Top