Question

Let \(F\) be a field. Show that in the ring \(F[X, Y],\) the ideal \((X, Y)\) is not a principal ideal.

Short answer

The ideal (X, Y) is not principal in the polynomial ring F[X, Y] because, if we assume it's a principal ideal generated by a polynomial P(X, Y), we encounter a contradiction in the degrees of the generator polynomial P(X, Y) and the elements X and Y. The degrees of P(X, Y), X, and Y do not satisfy the rules for polynomial multiplication in F[X, Y], showing that our assumption is false and (X, Y) cannot be a principal ideal in the polynomial ring F[X, Y].

Step-by-step solution

Assume ideal is principal

Let's assume that \((X,Y)\) is indeed a principal ideal generated by some non-zero polynomial \(P(X,Y) \in F[X,Y]\). So we can write \((X,Y) = (P)\), and every polynomial in \((X,Y)\) can be represented as a multiple of \(P\).

Express X and Y as multiples of P

Since \((X,Y)\) is the ideal generated by \(X\) and \(Y\), both \(X\) and \(Y\) are elements of the ideal. So, we should be able to express \(X\) and \(Y\) as multiples of \(P(X,Y)\), meaning there exist polynomials \(Q_1(X,Y), Q_2(X,Y)\in F[X,Y]\) such that \(X=P(X,Y)Q_1(X,Y)\) and \(Y=P(X,Y)Q_2(X,Y)\).

Evaluate the degrees

Since \(F[X,Y]\) is a polynomial ring and \(F\) is a field, we know that the degree of the product of two polynomials is equal to the sum of their degrees. Thus, \(\mathrm{deg}(X) = \mathrm{deg}(PQ_1) = \mathrm{deg}(P) + \mathrm{deg}(Q_1)\), and \(\mathrm{deg}(Y)= \mathrm{deg}(PQ_2)= \mathrm{deg}(P)+\mathrm{deg}(Q_2)\).

Degree contradiction

We know that \(\mathrm{deg}(X)=1\) and \(\mathrm{deg}(Y)=1\). Since the degrees of \(Q_1\) and \(Q_2\) must be non-negative, this implies that \(\mathrm{deg}(P)=0\), which means \(P(X,Y)\) would be a constant polynomial. However, this contradicts the assumption that \(P(X,Y)\) generates \((X,Y)\), as a constant polynomial cannot generate non-constant polynomials like \(X\) and \(Y\).

Conclusion

The contradiction in the degrees of the generator polynomial \(P(X,Y)\) and the elements \(X\) and \(Y\) shows that our assumption was false. Therefore, the ideal \((X, Y)\) cannot be a principal ideal in the polynomial ring \(F[X, Y]\).

Key concepts

Polynomial Ring

In algebra, a polynomial ring is a mathematical ring that consists of polynomials. Formally, if \(F\) is a field, the polynomial ring \(F[X, Y]\) consists of all polynomials in the variables \(X\) and \(Y\) with coefficients from the field \(F\). This means every element in the ring is a finite sum of terms of the form \(aX^mY^n\), where \(a \in F\) and \(m, n \geq 0\).
Polynomial rings are foundational structures in abstract algebra because they generalize arithmetic to polynomials. Important properties of polynomial rings include:

• The ring is commutative, meaning that the order of multiplication does not matter.
• It has a multiplicative identity, which is the constant polynomial 1.
• Polynomials can be added, subtracted, and multiplied, operations that satisfy the usual laws of algebra.

These properties make polynomial rings an essential tool for studying other algebraic structures and concepts.

Principal Ideal

The concept of a principal ideal is pivotal in ring theory. A principal ideal in a ring is an ideal that can be generated by a single element. This means every element in the ideal can be written as a product of a fixed element and some element from the ring.
For example, in a ring \(R\), an ideal \(I\) is said to be principal if there exists an element \(a \in R\) such that every element \(b \in I\) can be written in the form \(b = ra\), where \(r\) is an element in the ring.
Principal ideals are crucial because they offer a simpler representation of an ideal, making it easier to analyze and understand certain properties of the ring. However, not all rings have every ideal as principal, and showing that an ideal is not principal can reveal significant information about the ring's structure.

Degree of Polynomial

The degree of a polynomial is a key aspect in understanding its behavior and properties. For a given polynomial, the degree is the highest sum of exponents on its variables, considering each term independently.
For example, in the polynomial \(2X^3Y^2 + 3XY + 1\), the degree is 5, which arises from the term \(2X^3Y^2\) where \(3 + 2 = 5\). The degree is important because it provides insights into the polynomial's shape and solutions.
Some properties relating to degrees in polynomial rings are:

• The degree of a non-zero constant polynomial is 0.
• The degree of the sum of two polynomials is at most the maximum of their degrees.
• The degree of the product of two polynomials is the sum of their degrees.

Understanding the degree is crucial when analyzing whether an ideal in a polynomial ring is principal. If a polynomial intended to generate an ideal doesn't match the degree requirements to express specific elements, it can show contradictions, like in proving non-principal ideals.

Ideal in Algebra

An ideal in algebra is a subset of a ring that is closed under ring operations and absorbs multiplication by elements of the ring. In simpler terms, if you have a ring \(R\) and an ideal \(I\), then for every \(r \in R\) and \(i \in I\), the product \(ri\) is still in \(I\).
This absorption property is what distinguishes ideals from general subsets. Here are a few types of ideals:

• Proper Ideal: An ideal that is not equal to the entire ring.
• Principal Ideal: Generated by a single element, as discussed earlier.
• Maximal Ideal: An ideal that is not contained in any larger proper ideal.

In polynomial rings, ideals consist of all polynomial combinations such as the form \(X, Y\), which capture the structure and symmetry of the ring. Studying whether these ideals are principal or not helps in understanding the underlying algebraic framework better. This kind of questioning, like asking if \((X, Y)\) is principal in the polynomial ring \(F[X, Y]\), helps uncover deeper ring properties.