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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.
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참고문헌 (21)
참고문헌 신청
G. Artico / 1976 / On the compactness of minimal spectrum / Rend. Sem. Mat. Univ. Padova 56:79~84
G. Artico / 1981 / A subspace of Spec(A)and its connexions with the maximal ring of quotients / Rend. Sem. Mat. Univ. Padova 64:93~107
F. Azarpanah / 2002 / Algebraic characterizations of some discon-nected spaces / Ital. J. Pure Appl. Math. No 12:155~168
B. Banaschewski / 1988 / Another look at the localic Tychonoff theorem / Comment. Math. Univ. Carolin 29(4):647~656
B. Banaschewski / 1996 / Radical ideals and coherent frames / Comment. Math. Univ. Carolin 37(2):349~370
G. Artico / 1981 / A subspace of Spec(A)and its connexions with the maximal ring of quotients / Rend. Sem. Mat. Univ. Padova 64:93~107
F. Azarpanah / 2002 / Algebraic characterizations of some discon-nected spaces / Ital. J. Pure Appl. Math. No 12:155~168
B. Banaschewski / 1988 / Another look at the localic Tychonoff theorem / Comment. Math. Univ. Carolin 29(4):647~656
B. Banaschewski / 1996 / Radical ideals and coherent frames / Comment. Math. Univ. Carolin 37(2):349~370
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