Rings in which sums of $d$-Ideals are $d$-ideals
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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.
Abstract
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.
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