عنوان مجله
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Revista de la Real Academia de Ciencias Exactas Fisicas y Naturales Serie A-Matematicas
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چکیده
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In this paper, using the canonical correspondence between the idempotents and clopens,
we obtain several new results on lifting idempotents. The Zariski clopens of the maximal
spectrum are precisely determined, then as an application, lifting idempotents modulo the
Jacobson radical is characterized. Lifting idempotents modulo an arbitrary ideal is also characterized
in terms of certain connected sets related to that ideal. Then as an application, we
obtain that the sum of a lifting ideal and a regular ideal is a lifting ideal. We prove that
lifting idempotents preserves the orthogonality in countable cases. The lifting property of
an arbitrary morphism of rings is characterized. As another major result, it is proved that
the number of idempotents of a ring R is finite if and only if it is of the form 2κ where κ is
the cardinal of the connected components of Spec(R). Finally, it is proved that the primitive
idempotents of a zero dimensional ring are in 1-1 correspondence with the isolated points
of its prime spectrum. These results either generalize or improve several important results in
the literature.
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