Endomorphism Rings of Free Modules

Brodskii, G. M.

doi:10.1070/sm1974v023n02abeh002177

Cited by 4 publications

(4 citation statements)

References 16 publications

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“…The equivalence between conditions 1 and 2 was proved in [6]. The equivalence between conditions 5 and 6 follows from the fact that B 0 is essential in B A and hence…”

Section: Theoremmentioning

confidence: 94%

“…In [6] it is shown that R is QF if and only if A is left self-injective. In contrast, the ring B cannot be even left or right quasi-continuous [7] for any ring R. Our first result shows how to compute the injective hull of B B if R is QF.…”

Section: Lemma 2 a Left R-module M Is Quasi-injective If And Only Ifmentioning

confidence: 99%

“…At first sight it might seem that the rings A and B are too big to reflect properties of the ring R, and still more unexpected that A or B could encode finiteness conditions of R. However, already in [6,12] it is shown that the ring A reflects some finiteness conditions of R. There is a long tradition in the study of the ring theoretical properties of the ring A (among others see [2,6,9,12]). Recently several authors have shown interest in the study of the ring B (see, e.g., [5,7,10]).…”

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confidence: 99%

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Finiteness conditions and infinite matrix rings

Río

Simón

2005

Proc. Amer. Math. Soc.

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Abstract. For a unital ring R, RCFM α (R) denotes the ring of row and column finite matrices over R indexed by α. We give necessary and sufficient structural conditions on RCFM α (R) which are equivalent to R being, respectively, Quasi-Frobenius, left artinian, and left noetherian.In this paper R denotes an (associative and unital) ring and α is an infinite set. We use the following notation, where "matrices" means "matrices indexed by α with entries in R":(1) A = RFM α (R) = ring of row finite matrices, B = RCFM α (R) = ring of row and column finite matrices, B 0 = FM α (R) = ring of finite matrices.At first sight it might seem that the rings A and B are too big to reflect properties of the ring R, and still more unexpected that A or B could encode finiteness conditions of R. However, already in [6,12] it is shown that the ring A reflects some finiteness conditions of R. There is a long tradition in the study of the ring theoretical properties of the ring A (among others see [2,6,9,12]). Recently several authors have shown interest in the study of the ring B (see, e.g., [5,7,10]). In this paper we study the properties of the ring B under the assumption that R satisfies some finiteness condition (quasi-Frobenius, artinian, noetherian). The relationship between a ring R and A comes essentially from the adjoint pair Hom R (F, −) : R − mod A − mod : F ⊗ A −, where F is a free left R-module of rank |α| and A is canonically identified with End R (F ). An interesting exception may be found in the computation of the Jacobson radical (see [13]), where the amount of matrix manipulation exceeds adjunction techniques. If one wants to relate the rings R and B one can also use the adjoint pair Hom R (F, −) :The difference in the performance of the adjoint pairs for A and B relies on the fact that while A is in the image of Hom R (F, −), namely A End R (F ), this is not the case for B. However, we may still use some "adjoint-like" techniques to relate some special objects in the category of R-modules and some special matrices or ideals of B and keeping in mind that B is the ring of continuous endomorphisms of F in a certain topology [11].We start with some notation. If i, j ∈ α, then e ij denotes the element of B 0 having 1 in the (i, j)-th entry and zeroes elsewhere, and if a ∈ A, then a(i, j) denotes the (i, j)-th entry of a. Set e i = e ii = e {i} . If F is a subset of α, then

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“…The equivalence between conditions 1 and 2 was proved in [6]. The equivalence between conditions 5 and 6 follows from the fact that B 0 is essential in B A and hence…”

Section: Theoremmentioning

confidence: 94%

Section: Lemma 2 a Left R-module M Is Quasi-injective If And Only Ifmentioning

confidence: 99%

mentioning

confidence: 99%

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Finiteness conditions and infinite matrix rings

Río

Simón

2005

Proc. Amer. Math. Soc.

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“…Finally we mention that special cases of (1) which are solvable by elementary methods arose as mathematical competition problems for secondary school students [10,18]. In this setting, F is linear in its first r arguments, but the coefficients are allowed to be (known) functions of t. We illustrate this and the general idea of the method via the subsequent examples.…”

Section: Introductionmentioning

confidence: 99%