A note on Noetherian orders in Artinian rings

Chatters, A. W.

doi:10.1017/s0017089500003827

Cited by 5 publications

(3 citation statements)

References 10 publications

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“…We shall show that N = NP. Suppose that e is an idempotent element of R such that eN is a non-zero serial right R-module and eNP = eN 2 ; we shall obtain a contradiction. We have Kdim(eN/eNP)^Kdim(N)<Kdim(R) = Kdim(R/P).…”

Section: Let R Be An Indecomposable Serial Ring With Krull Dimension mentioning

confidence: 94%

“…Suppose that eN^O and that eNi=eNP. By taking K = eN/eN 2 in Lemma 3.1, we see that eNP-eN 2 . Because X is not contained in P, there is no minimal prime of R which contains X + P. Therefore Proof.…”

Section: Let R Be An Indecomposable Serial Ring With Krull Dimension mentioning

confidence: 97%

“…Let y eeN with y + eNP = x. Then y(cR + P) c eN 2 Proof. There are orthogonal primitive idempotents g x ,.…”

Section: Let R Be An Indecomposable Serial Ring With Krull Dimension mentioning

confidence: 99%

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Serial rings with krull dimension

Chatters¹

1990

Glasgow Math. J.

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A module is said to be serial if it has a unique chain of submodules, and a ring is serial if it is a direct sum of serial right ideals and a direct sum of serial left ideals. The serial rings of Krull dimension 0 are the Artinian serial (or generalised uniserial) rings studied by Nakayama and for which there is an extensive theory (see for example [4]). Warfield in [10] extended the theory to the non-Artinian case. In particular he showed that a Noetherian serial ring is a direct sum of Artinian serial rings and prime Noetherian serial rings, and he gave a structure theorem in the prime Noetherian case. A Noetherian non-Artinian serial ring has Krull dimension 1. Serial rings of arbitrary Krull dimension have been studied by Wright ([9], [12], [13], [14]) with special results being proved when the Krull dimension is 1 or 2.

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Section: Let R Be An Indecomposable Serial Ring With Krull Dimension mentioning

confidence: 94%

Section: Let R Be An Indecomposable Serial Ring With Krull Dimension mentioning

confidence: 97%

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Serial rings with krull dimension

Chatters¹

1990

Glasgow Math. J.

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Modules

Markov¹,

Mikhalëv²,

Skornyakov³

et al. 1983

J Math Sci

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Rings generated by their regular elements

Chatters¹,

Ginn

1984

Glasgow Math. J.

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1. Introduction. The units of a ring R are defined by means of a multiplicative property, but in many cases they generate R additively. For example, it is shown in [5, Proposition 6] that if R is a semi-simple Artinian ring then every element of R is a sum of units if and only if the ring S = Z/2Z©Z/2Z is not a direct summand of R, where Z denotes the ring of integers. The theme of this paper is to investigate the corresponding situation concerning regular elements, i.e. elements which are not zero-divisors. We show that if R is a semi-prime right Goldie ring then every element of R is a sum of regular elements if and only if R does not have the ring S defined above as a direct summand (Corollary 2.9). We also characterise those Noetherian rings R such that every element of R is a sum of regular elements (Theorem 2.6). The characterisation is in terms of the nature of certain prime factor rings of R, and it is again the presence of the ring S, this time in a particular way as a factor ring of R, which prevents R from being generated by its regular elements. If R has no non-zero Artinian one-sided ideals or if 2 is a regular element of R, then every element of R is a sum of regular elements (Corollaries 2.5 and 2.7). As an application we show in Section 3 that, for many Noetherian rings R, the set of elements of R which are divisible by every regular element of R is a two-sided ideal of R.

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A note on Noetherian orders in Artinian rings

Cited by 5 publications

References 10 publications

Serial rings with krull dimension

Serial rings with krull dimension

Modules

Rings generated by their regular elements

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