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On $α$-regular archimedean $f$-rings

In: Mathematica Slovaca, vol. 61, no. 3
Anthony W. Hager - Jorge Martínez
Detaily:
Rok, strany: 2011, 469 - 484
Kľúčové slová:
$\alpha$-regular, archimedean $f$-rings, bounded inversion, hull classes, rings of quotients
O článku:
For a commutative ring $A$ with identity, and for infinite cardinals $α$ as well as the symbol $∞$, which indicates the situation in which there are no cardinal restrictions, one defines $A$ to be $α$-regular if for each subset $D$ of $A$, with $|D|<α$ and $de=0$, for any two distinct $d,e\in D$, there is an $s\in A$ such that $d2s=d$, for each $d\in D$, and if $xd=0$, for each $d\in D$, then $xs=0$, This paper studies $α$-regular archimedean $f$-rings, relative to lateral $α$-completeness. The main result is that the operator $l(α)$ that gives the lateral $α$-completion commutes with $b$, the reflection that closes an $f$-ring with respect to bounded inversion. An $f$-ring is $α$-regular if and only if it has bounded inversion and is laterally $α$-complete, and the operator that creates the $α$-regular hull is $r(α)=b· l(α)$. It is shown that the space $mr(α)A$ of all maximal $\ell$-ideals of $r(α)A$ is the same as that of the $α$-projectable hull. Finally, $r(α)A$ contains the ring of $α$-quotients, and necessary conditions are given for them to coincide.
Ako citovať:
ISO 690:
Hager, A., Martínez, J. 2011. On $α$-regular archimedean $f$-rings. In Mathematica Slovaca, vol. 61, no.3, pp. 469-484. 0139-9918. DOI: https://doi.org/10.2478/s12175-011-0024-3

APA:
Hager, A., Martínez, J. (2011). On $α$-regular archimedean $f$-rings. Mathematica Slovaca, 61(3), 469-484. 0139-9918. DOI: https://doi.org/10.2478/s12175-011-0024-3
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