Understanding the Irregularity of the Commutative Ring (frac{mathbb{C}[x, y]}{(x^2, y^2)})

When delving into the intricate world of commutative algebra, one often comes across rings with peculiar and non-intuitive properties. A notable example is the quotient ring (frac{mathbb{C}[x, y]}{(x^2, y^2)}), which is neither a regular ring, von Neumann regular, nor homologically regular. This article will explore the reasons behind its irregularity and its implications within the broader context of abstract algebra.

Introduction and Context

The polynomial ring (mathbb{C}[x, y]) consists of all polynomials in two indeterminates (x) and (y) with complex coefficients. When we quotient this ring by the ideal generated by (x^2) and (y^2), we obtain the ring (frac{mathbb{C}[x, y]}{(x^2, y^2)}). This quotient ring has elements of the form [a bx cy dxy,] where (a, b, c, d in mathbb{C}). The ring (frac{mathbb{C}[x, y]}{(x^2, y^2)}) thus contains the nilpotent elements (x) and (y), with (x^2 y^2 0).

Why is (frac{mathbb{C}[x, y]}{(x^2, y^2)}) NOT a Regular Ring?

A regular ring is a commutative ring in which every localization at a prime ideal is a regular local ring. A regular local ring requires that the maximal ideal is generated by a regular sequence. In our context, the ring (frac{mathbb{C}[x, y]}{(x^2, y^2)}) is not a regular ring because it does not satisfy this condition.

Lack of a Regular Local Ring Property

Consider the localization of the ring (frac{mathbb{C}[x, y]}{(x^2, y^2)}) at the maximal ideal ((x, y)). This localization results in a ring that still contains the nilpotent elements (x) and (y) since (x^2 y^2 0). This means that the localization is not a domain, as it contains zero divisors. Regular local rings, by definition, must be domains. Therefore, (frac{mathbb{C}[x, y]}{(x^2, y^2)}) is not a regular ring because its localization at the maximal ideal ((x, y)) is not a domain.

Dimension and Irregularity

The Krull dimension of (frac{mathbb{C}[x, y]}{(x^2, y^2)}) is one, since the maximal ideal ((x, y)) is generated by two elements. However, a regular ring with dimension one must be a discrete valuation ring or a regular local ring. Since (frac{mathbb{C}[x, y]}{(x^2, y^2)}) is not a domain, it cannot be regular.

Why is (frac{mathbb{C}[x, y]}{(x^2, y^2)}) NOT von Neumann Regular?

A von Neumann regular ring is a ring in which for every element (a) in the ring, there exists an element (b) such that (a aba). In (frac{mathbb{C}[x, y]}{(x^2, y^2)}), the elements (x) and (y) are nilpotent, meaning that (x^2 y^2 0). These nilpotent elements do not satisfy the conditions for being von Neumann regular because there is no element (b) in the ring such that (x xbx) or (y yby).

Why is (frac{mathbb{C}[x, y]}{(x^2, y^2)}) NOT Homologically Regular?

A homologically regular ring is a ring that is locally regular. This means that every localization of the ring at a prime ideal must be a regular local ring. Since (frac{mathbb{C}[x, y]}{(x^2, y^2)}) is not a regular ring due to the reasons mentioned earlier, it cannot be homologically regular. In particular, localizing at the maximal ideal ((x, y)) results in a ring with zero divisors, which means it is not a regular local ring.

Conclusion

The quotient ring (frac{mathbb{C}[x, y]}{(x^2, y^2)}) is a prime example of a ring that showcases the intricacies and irregularities that can arise in the study of abstract algebra. While it possesses certain algebraic properties, it fails to meet the criteria for being a regular ring, von Neumann regular, or homologically regular. These irregularities enrich our understanding of the behavior of rings and the importance of carefully defining and verifying these properties in various contexts.

Keywords: regular ring, commutative ring, von Neumann regular, Krull dimension