3,446 research outputs found
Properties of chains of prime ideals in an amalgamated algebra along an ideal
Let be a ring homomorphism and let be an ideal of . In
this paper, we study the amalgamation of with along with respect to
(denoted by ), a construction that provides a general frame
for studying the amalgamated duplication of a ring along an ideal, introduced
and studied by D'Anna and Fontana in 2007, and other classical constructions
(such as the , the and the constructions). In
particular, we completely describe the prime spectrum of the amalgamated
duplication and we give bounds for its Krull dimension.Comment: J. Pure Appl. Algebra (to appear
The patch topology and the ultrafilter topology on the prime spectrum of a commutative ring
Let R be a commutative ring and let Spec(R) denote the collection of prime
ideals of R. We define a topology on Spec(R) by using ultrafilters and
demonstrate that this topology is identical to the well known patch or
constructible topology. The proof is accomplished by use of a von Neumann
regular ring canonically associated with .Comment: A Remark was added at the end of the paper. To appear in Comm.
Algebr
Uppers to zero and semistar operations in polynomial rings
Given a stable semistar operation of finite type on an integral
domain , we show that it is possible to define in a canonical way a stable
semistar operation of finite type on the polynomial ring , such
that is a -quasi-Pr\"ufer domain if and only if each upper to zero
in is a quasi--maximal ideal. This result completes the
investigation initiated by Houston-Malik-Mott \cite[Section 2]{hmm} in the star
operation setting. Moreover, we show that is a Pr\"ufer
-multiplication (resp., a -Noetherian; a -Dedekind) domain
if and only if is a Pr\"ufer -multiplication (resp., a
-Noetherian; a -Dedekind) domain. As an application of the
techniques introduced here, we obtain a new interpretation of the
Gabriel-Popescu localizing systems of finite type on an integral domain
(Problem 45 of \cite{cg}), in terms of multiplicatively closed sets of the
polynomial ring
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 \cite{G}) and, hence, the related classical theory of ideal
systems based on the works by W. Krull, E. Noether, H. Pr\"{u}fer and P.
Lorenzen from 1930's. In \cite{FL1} and \cite{FL2} the current authors
investigated properties of the Kronecker function rings which arise from
arbitrary semistar operations on an integral domain . In this paper we
extend that study and also generalize Kang's notion of a star Nagata ring
\cite{Kang:1987} and \cite{Kang:1989} to the semistar setting. Our principal
focuses are the similarities between the ideal structure of the Nagata and
Kronecker semistar rings and between the natural semistar operations that these
two types of function rings give rise to on .Comment: 20 page
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