Strong Separativity over Regular Rings
An ideal I of a ring R is strongly separative provided that for all finitely generated projective R-modules A, B with A=AI and B=BI, if 2A ≅ A ⊕ B, then A ≅ B. We prove in this paper that a regular ideal I of a ring R is strongly separative if and only if each a ∈ 1+I satisfying (1-a)R ∝ r(a) is unit-regular, if and only if each a ∈ 1+I satisfying (1-a2)R ∝ r(a2) is unit-regular, if and only if each a ∈ 1+I satisfying R(1-a)R=R r(a) is unit-regular, if and only if each a ∈ 1+I satisfying R(1-a2)R=R r(a2) is unit-regular.
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1971 ◽
Vol 4
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pp. 57-62
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1989 ◽
Vol 32
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pp. 333-339
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1969 ◽
Vol 12
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pp. 417-426
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1982 ◽
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2018 ◽
Vol 55
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pp. 270-279
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