J. Korean Math. Soc. 2019; 56(2): 539-558
Online first article August 6, 2018 Printed March 1, 2019
https://doi.org/10.4134/JKMS.j180277
Copyright © The Korean Mathematical Society.
Themba Dube
University of South Africa
An ideal of a commutative ring is called a $d$-ideal if it contains the annihilator of the annihilator of each of its elements. Denote by DId$(A)$ the lattice of $d$-ideals of a ring $A$. We prove that, as in the case of $f$-rings, DId$(A)$ is an algebraic frame. Call a ring homomorphism ``compatible'' if it maps equally annihilated elements in its domain to equally annihilated elements in the codomain. Denote by $\mathbf{SdRng}_\mathrm{c}$ the category whose objects are rings in which the sum of two $d$-ideals is a $d$-ideal, and whose morphisms are compatible ring homomorphisms. We show that DId$\colon\mathbf{SdRng}_{\mathrm{c}}\to\mathbf{CohFrm}$ is a functor ($\mathbf{CohFrm}$ is the category of coherent frames with coherent maps), and we construct a natural transformation RId$\longrightarrow$DId, in a most natural way, where RId is the functor that sends a ring to its frame of radical ideals. We prove that a ring $A$ is a Baer ring if and only if it belongs to the category $\mathbf{SdRng}_{\mathrm{c}}$ and DId$(A)$ is isomorphic to the frame of ideals of the Boolean algebra of idempotents of $A$. We end by showing that the category $\mathbf{SdRng}_{\mathrm{c}}$ has finite products.
Keywords: Baer ring, reduced ring, $d$-ideal, sum of $d$-ideals, $\zeta$-ideal, algebraic frame, $d$-nucleus, functor
MSC numbers: 13A15, 06D22, 18A05
Supported by: The author acknowledges funding from the National Research Foundation of South Africa through a research grant with grant number 113829.
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