On Direct Sums of Extending Modules and Internal Exchange Property

Hanada, Katutosi; Kuratomi, Yosuke; Oshiro, Kiyoichi

doi:10.1006/jabr.2001.9089

Cited by 18 publications

(4 citation statements)

References 11 publications

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“…The internal exchange property is used by Hanada, Kuratomi and Oshiro in [143] and [144] and by Mohammed and Müller in [244] in their investigations into the thorny question of when a direct sum of extending modules is extending. The basic features of the property given in 11.36 to 11.40 appear in [244] and [245] and these and exchange decompositions are used significantly in Chapter 4.…”

Section: Comments As Already Indicated the Exchange Property Was Inmentioning

confidence: 99%

“…Conversely, suppose N is an amply supplemented module such that N = Re [144,Theorem 2.11]). Generalised relative injectivity was renamed as ojective by Mohamed and Müller (in [244]) in honour of Oshiro.…”

Section: Examplementioning

confidence: 99%

“…[61]; Hanada, Kado and Oshiro [143]; Kuratomi [214]; Kutami and Oshiro [215]; Miyashita [235]; Mohamed, Müller and Singh [246]; Orhan and Keskin Tütüncü [268]. ; As lifting modules and self-projective modules are closed under direct summands, any direct summand of a strongly discrete module is strongly discrete.…”

Section: Factors Of Quasi-discrete Modules Suppose M Is a Quasi-discmentioning

confidence: 99%

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Lifting Modules

2006

Frontiers in Mathematics

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Section: Comments As Already Indicated the Exchange Property Was Inmentioning

confidence: 99%

Section: Examplementioning

confidence: 99%

Section: Factors Of Quasi-discrete Modules Suppose M Is a Quasi-discmentioning

confidence: 99%

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Lifting Modules

2006

Frontiers in Mathematics

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“…In 1988, Kamal and Müller, see [8], [9], [10] studied the concept of extending modules over Noetherian rings and commutative domains. Hanada, Kuratomi and Oshiro [6] studied the concept of extending modules. The following open problem was posed by Harmanci and Smith [7].…”

Section: Introductionmentioning

confidence: 99%

Ojective ideals in modular lattices

Nimbhorkar

Shroff

2015

Czech Math J

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Decompositions of modules

Lifting Modules

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The exchange property appears in rudimentary form as the Steinitz Exchange Lemma for vector spaces (see, for example, [73, Lemma 4.3.4]) and, as a slightly more general forerunner, in the following well-known property of semisimple modules (see [363, 20.1]). Steinitz' Exchange Lemma for semisimple modules. LetIn module theory, the exchange property plays a prominent rôle in the study of isomorphic refinements of direct sum decompositions of modules. We first record two elementary lemmas on submodules of direct sums which will be frequently useful. LemmaProof. Since M 1 ⊆ N , the modularity property gives11.3. Lemma. Let M = M 1 ⊕ M 2 , let π 1 : M → M 1 be the projection map, and let N be a submodule of M . Then M = N ⊕ M 2 if and only if the restriction π 1 | N : N → M 1 is an isomorphism. In this case, the projection map π N : M → N associated with the direct sum M = N ⊕ M 2 is given by π 1 (π 1 | N ) −1 . 11.7. Corollary. Let c be a cardinal number and I be an index set of cardinality at most c. Let A, M, N, L, and A i , i ∈ I, be modules such thatIf M has the c-exchange property, then there exist submodules B i ⊆ A i such thatProof. This follows from the Lemma after factoring out L and applying the cexchange property. Direct sums and thec-exchange property. Let c be a cardinal number and let M = X ⊕ Y . Then M has the c-exchange property if and only if both X and Y have the c-exchange property. Proof. Assume that both X and Y have the c-exchange property and let A = M ⊕ N = I A i , where card(I) ≤ c. Then, by our assumption on Y ,This implies that A = X ⊕ ( I B i ). Hence X has the c-exchange property.As a corollary we obtain: 11.9. Finite direct sums and the exchange properties. Let M = M 1 ⊕ · · · ⊕ M k . Then M has the (finite) exchange property if and only if M i has the (finite) exchange property for each i = 1, . . . , k.Example 12.15 will show that 11.9 does not generalise to infinite direct sums. Proposition. LetIf T has the finite exchange property and N ⊆ L, then K has the finite exchange property.a direct summand of T . Hence K has the finite exchange property. Chapter 3. Decompositions of modules 11.12. Summable homomorphisms. Let M and N be R-modules. We say that a family {f i : M → N } I of homomorphisms is summable if, given any m ∈ M , then (m)f i = 0 for all but a finite number of indices i ∈ I. In this case we can define the sum I f i of the family, in the obvious way. Clearly, if {f i : M → N } I is summable, so too is {f i g i : M → K} i∈I for any R-module K and any family of homomorphisms {g i : N → K} I . The following result shows that for an R-module M it suffices to check the exchange property in the case when M is isomorphic to a direct sum of clones of itself. Moreover, it also characterises the exchange property using summable families in M 's endomorphism ring. 11.13. The exchange property and summable families. Let M be an R-module, S = End(M ), and c be any cardinal. Then the following statements are equivalent. (a) M has the c-exchange property; (b) for any index set I with car...

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On Direct Sums of Extending Modules and Internal Exchange Property

Cited by 18 publications

References 11 publications

Lifting Modules

Lifting Modules

Ojective ideals in modular lattices

Decompositions of modules

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