Kronecker function rings and abstract Riemann surfaces

A Generalization of Kronecker Function Rings and Nagata Rings

Let D be an integral domain with quotient field K. The Nagata ring D(X) and the Kronecker function ring Kr(D) are both subrings of the field of rational functions K(X) containing as a subring the ring D[X] of polynomials in the variable X. Both of these function rings have been extensively studied and generalized. The principal interest in these two extensions ofD lies in the reflection of vari...

Nagata Rings, Kronecker Function Rings and Related Semistar Operations

In 1994, Matsuda and Okabe introduced the notion of semistar operation. This concept extends the classical concept of star operation (cf. for instance, Gilmer’s book [20]) and, hence, the related classical theory of ideal systems based on the works by W. Krull, E. Noether, H. Prüfer and P. Lorenzen from 1930’s. In [17] and [18] the current authors investigated properties of the Kronecker functi...

An historical overview of Kronecker function rings, Nagata rings, and related star and semistar operations

Riemann Surfaces

Riemann introduced his surfaces in the middle of the 19th century in order to “geometrize” complex analysis. In doing so, he paved the way for a great deal of modern mathematics such as algebraic geometry, manifold theory, and topology. So this would certainly be of interest to students in these areas, as well as in complex analysis or number theory. In simple terms, a Riemann surface is a surf...

Crystallography and Riemann Surfaces

The level set of an elliptic function is a doubly periodic point set in C. To obtain a wider spectrum of point sets, we consider, more generally, a Riemann surface S immersed in C2 and its sections (“cuts”) by C. We give S a crystallographic isometry in C2 by defining a fundamental surface element as a conformal map of triangular domains and S as its extension by reflections in the triangle edg...