Narrow Results

Let $R={\oplus }_{\alpha \in \mathrm{\Gamma }}{R}_{\alpha }$ be an integral domain graded by a nontrivial torsionless grading monoid $\mathrm{\Gamma }$. Among other things, we show that if $\mathrm{\Gamma }\cap (-\mathrm{\Gamma })=\{0\}$, then every nonzero homogeneous ideal of $R$ is invertible if and only if $R_0$ is a field and $R\cong R_0[X]$, where $X$ is an indeterminate over $R_0$, if and only if $R$ is a Dedekind domain, if and only if $R$ is a PID.

Starting from the notion of semistar operation, introduced in 1994 by Okabe and Matsuda [49], which generalizes the classical concept of star operation (cf. Gilmer's book [27]) and, hence, the related classical theory of ideal systems based on the works by W. Krull, E. Noether, H. Prüfer, P. Lorenzen and P. Jaffard (cf. Halter–Koch's book [32]), in this paper we outline a general approach to the theory of Prüfer ⋆-multiplication domains (or P⋆MDs), where ⋆ is a semistar operation. This approach leads to relax the classical restriction on the base domain, which is not necessarily integrally closed in the semistar case, and to determine a semistar invariant character for this important class of multiplicative domains (cf. also J. M. García, P. Jara and E. Santos [25]). We give a characterization theorem of these domains in terms of Kronecker function rings and Nagata rings associated naturally to the given semistar operation, generalizing previous results by J. Arnold and J. Brewer ]10] and B. G. Kang [39]. We prove a characterization of a P⋆MD, when ⋆ is a semistar operation, in terms of polynomials (by using the classical characterization of Prüfer domains, in terms of polynomials given by R. Gilmer and J. Hoffman [28], as a model), extending a result proved in the star case by E. Houston, S. J. Malik and J. Mott [36]. We also deal with the preservation of the P⋆MD property by ascent and descent in case of field extensions. In this context, we generalize to the P⋆MD case some classical results concerning Prüfer domains and PvMDs. In particular, we reobtain as a particular case a result due to H. Prüfer [51] and W. Krull [41] (cf. also F. Lucius [43] and F. Halter-Koch [34]). Finally, we develop several examples and applications when ⋆ is a (semi)star given explicitly (e.g. we consider the case of the standardv-, t-, b-, w-operations or the case of semistar operations associated to appropriate families of overrings).

A natural development of the recent work about semistar operations leads to investigate the concept of “semistar invertibility”. Here, we show the existence of a “theoretical obstruction” for extending many results, proved for star-invertibility, to the semistar case. For this reason, we introduce two distinct notions of invertibility in the semistar setting (called ⋆-invertibility and quasi-⋆-invertibility), we discuss the motivations of these “two levels” of invertibility and we extend, accordingly, many classical results proved for the d-, v-, t- and w-invertibility.

Let R be a commutative Noetherian Nagata ring, let M be a non-zero finitely generated R-module, and let I be an ideal of R such that height_MI > 0. In this paper, there is a definition of the integral closure N_a for any submodule N of M extending Rees' definition for the case of a domain. As the main results, it is shown that the operation N → N_a on the set of submodules N of M is a semi-prime operation, and for any submodule N of M, the sequences Ass_R M/(IⁿN)_a and Ass_R (IⁿM)_a/(IⁿN)_a(n=1,2,…) of associated prime ideals are increasing and ultimately constant for large n.

Let $R$ and $S$ be two commutative rings with unity, let $J$ be an ideal of $S$ and $\phi :R\to S$ be a ring homomorphism. In this paper, we give a characterization for the amalgamated algebra $R{\bowtie }^{\phi }J$ to be a Nagata ring, a strong S-domain, and a catenarian. Also, we investigate the conditions that the ring of Hurwitz series over $R{\bowtie }^{\phi }J$ has a complete comaximal factorization.