Exchange Rings and Modules
Let τ be a cardinal number. A module M is called a module with the τ-exchange property (see ) if for every module X and each direct decomposition X = M′⊕Y = ⊕ i∈I N i such that M′≌M and card(I) ≤ τ, there are submodules N i ′⊆N i (i ∈ I) with X = M′⊕(⊕ i∈I N i ′. (It follows from the modular law that N i ′ must be a direct summand of N i for all i.)
KeywordsDirect Summand Exchange Ring Exchange Module Invertible Element Regular Ring
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