Proving I2 is a Principal Ideal while I is not in the Ring R : ?[x, y] / (x2 y2 - 1)

In the study of commutative algebra, rings and ideals play a pivotal role. This article will explore the properties of a specific ideal I in the ring R : ?[x, y] / (x2 y2 - 1). We will delve into why I2 is a principal ideal, while I itself is not.

1. Introduction to Commutative Algebra and Ring Theory

Commutative algebra is an essential branch of algebra that studies commutative rings and their ideals. A ring is a set equipped with two operations: addition and multiplication. The ring ?[x, y] / (x2 y2 - 1) is a quotient ring, which means it is formed by taking the polynomial ring ?[x, y] and considering the equivalence classes under a specific ideal generated by the polynomial x2 y2 - 1.

2. Defining the Ring and Ideal

We define the ring R ?[x, y] / (x2 y2 - 1). Here, x2 y2 - 1 is the polynomial that defines the ideal (x2 y2 - 1). Each element in this ring can be thought of as a polynomial in x and y modulo x2 y2 - 1. For instance, any polynomial p(x, y) in ?[x, y] can be reduced modulo x2 y2 - 1 to give a unique representative in R.

Let I be the ideal generated by the element 1 in R. In this context, I is the set of all polynomial elements in R that can be written as a product of 1 with any other element in R.

3. Proving I2 is a Principal Ideal

The first step to proving that I2 is a principal ideal in R is to determine what I2 looks like. Since I is generated by 1, the square I2 is generated by the product of any two elements in I. In this case, the only element in I is the polynomial 1, so I2 consists of the polynomial 1 itself.

Formally, I2 (1), which is clearly a principal ideal generated by the element 1. Hence, we have shown that I2 is a principal ideal in R.

4. Proving I is not a Principal Ideal

Next, we explore why the ideal I is not a principal ideal in R. The ideal I is the set of all elements in R that multiply with 1. This includes the polynomial 1 itself and potentially other multiples of 1. However, the ideals in R are not determined by a single element, and we need to show that no single element can generate the entire set I.

Assume, for contradiction, that I is a principal ideal generated by some element g in R. Then, I (g) for some g in R. This would mean that every element in I can be written as a product of g with some element in R.

However, consider the element 1 in I. If I (g), then there must exist an element r in R such that 1 rg. But in ?[x, y] / (x2 y2 - 1), this is impossible because 1 is the multiplicative identity, and no polynomial g in R can have a multiplicative inverse that is also a polynomial in R when considering the equivalence classes modulo x2 y2 - 1. Therefore, I cannot be generated by a single element, and thus I is not a principal ideal in R.

5. Conclusion

In conclusion, we have demonstrated that in the ring R ?[x, y] / (x2 y2 - 1): I2 is a principal ideal generated by the element 1. I is not a principal ideal because no single element can generate the entire set I. These results highlight the intricate nature of ideals in quotient rings and provide insight into commutative algebra at a deeper level.

6. Additional Resources and Further Reading

For a more in-depth understanding of these concepts, consider consulting the following resources:

Commutative Algebra: with a View Toward Algebraic Geometry by David Eisenbud Rings, Modules, and Algebras in Stable Homotopy Theory by Anthony D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May Algebraic Topology by Allen Hatcher (Chapter 3, dealing with Homology Theory)

Understanding the properties of ideals in rings such as R is crucial for advanced studies in algebraic geometry, number theory, and other related fields.