On generalized Kronecker function rings, II

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...

Generalized GCD Rings II

Greatest common divisors and least common multiples of quotients of elements of integral domains have been investigated by Lüneburg and further by Jäger. In this paper we extend these results to invertible fractional ideals. We also lift our earlier study of the greatest common divisor and least common multiple of finitely generated faithful multiplication ideals to finitely generated projectiv...

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

On generalized AIP-rings

In this paper, we introduce the concept of the generalized AIP-rings as a generalization of the generalized quasiBaer rings and generalized p.p.-rings. We show that the class of the generalized AIP-rings is closed under direct products and Morita invariance. We also characterize the 2-by-2 formal upper triangular matrix rings of this new class of rings. Finally, we provide several examples to s...