Abstract
We investigate partial cancellation of modules and show that if an ideal I of an exchange ring R has stable range one, then A⊕B≌A⊕C implies B≌C for all A∈FP (I). The converse is true when R is a regular ring. For an ideal I of a regular ring, we also show that I has stable range one if and only if perspectivity is transitive in L(A) for all A∈ FP (I). These give nontrivial generalizations for unit-regularity.
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Chen, H. Partial cancellation of modules. Acta Mathematica Hungarica 100, 205–214 (2003). https://doi.org/10.1023/A:1025089309075
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DOI: https://doi.org/10.1023/A:1025089309075