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Search: MSC category 13A15 ( Ideals; multiplicative ideal theory )

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1. CMB 2015 (vol 59 pp. 197)

Rajaee, Saeed
Quasi-copure Submodules
All rings are commutative with identity and all modules are unital. In this paper we introduce the concept of quasi-copure submodule of a multiplication $R$-module $M$ and will give some results of them. We give some properties of tensor product of finitely generated faithful multiplication modules.

Keywords:multiplication module, arithmetical ring, copure submodule, radical of submodules
Categories:13A15, 13C05, 13C13, , 13C99

2. CMB 2008 (vol 51 pp. 406)

Mimouni, Abdeslam
Condensed and Strongly Condensed Domains
This paper deals with the concepts of condensed and strongly condensed domains. By definition, an integral domain $R$ is condensed (resp. strongly condensed) if each pair of ideals $I$ and $J$ of $R$, $IJ=\{ab/a \in I, b \in J\}$ (resp. $IJ=aJ$ for some $a \in I$ or $IJ=Ib$ for some $b \in J$). More precisely, we investigate the ideal theory of condensed and strongly condensed domains in Noetherian-like settings, especially Mori and strong Mori domains and the transfer of these concepts to pullbacks.

Categories:13G05, 13A15, 13F05, 13E05

3. CMB 2003 (vol 46 pp. 3)

Anderson, D. D.; Dumitrescu, Tiberiu
Condensed Domains
An integral domain $D$ with identity is condensed (resp., strongly condensed) if for each pair of ideals $I$, $J$ of $D$, $IJ=\{ij; i\in I, j\in J\}$ (resp., $IJ=iJ$ for some $i\in I$ or $IJ =Ij$ for some $j\in J$). We show that for a Noetherian domain $D$, $D$ is condensed if and only if $\Pic(D)=0$ and $D$ is locally condensed, while a local domain is strongly condensed if and only if it has the two-generator property. An integrally closed domain $D$ is strongly condensed if and only if $D$ is a B\'{e}zout generalized Dedekind domain with at most one maximal ideal of height greater than one. We give a number of equivalencies for a local domain with finite integral closure to be strongly condensed. Finally, we show that for a field extension $k\subseteq K$, the domain $D=k+XK[[X]]$ is condensed if and only if $[K:k]\leq 2$ or $[K:k]=3$ and each degree-two polynomial in $k[X]$ splits over $k$, while $D$ is strongly condensed if and only if $[K:k] \leq 2$.

Categories:13A15, 13B22

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