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On modules and rings with the restricted minimum condition

100%

Koşan M., Žemlička J.

Colloquium Mathematicum

2015

tom 140

nr 1

75-86

A module M satisfies the restricted minimum condition if M/N is artinian for every essential submodule N of M. A ring R is called a right RM-ring whenever $R_{R}$ satisfies the restricted minimum condition as a right module. We give several structural necessary conditions for particular classes of RM-rings. Furthermore, a commutative ring R is proved to be an RM-ring if and only if R/Soc(R) is noetherian and every singular module is semiartinian.

An intermediate ring between a polynomial ring and a power series ring

81%

Koşan M., Lee T.-K., Zhou Y.

Colloquium Mathematicum

2013

tom 130

nr 1

1-17

Let R[x] and R[[x]] respectively denote the ring of polynomials and the ring of power series in one indeterminate x over a ring R. For an ideal I of R, denote by [R;I][x] the following subring of R[[x]]: [R;I][x]: = {$∑_{i≥0} r_i x^i ∈ R[[x]]$ : ∃ 0 ≤ n∈ ℤ such that $r_i∈ I$, ∀ i ≥ n}. The polynomial and power series rings over R are extreme cases where I = 0 or R, but there are ideals I such that neither R[x] nor R[[x]] is isomorphic to [R;I][x]. The results characterizing polynomial rings or power series rings with a certain ring property suggest a similar study to be carried out for the ring [R;I][x]. In this paper, we characterize when the ring [R;I][x] is semipotent, left Noetherian, left quasi-duo, principal left ideal, quasi-Baer, or left p.q.-Baer. New examples of these rings can be given by specializing to some particular ideals I, and some known results on polynomial rings and power series rings are corollaries of our formulations upon letting I = 0 or R.

Ograniczanie wyników

2 Colloquium Mathematicum

2 Koşan M.

1 Lee T.-K.

1 Zhou Y.

1 Žemlička J.

1 2015

1 2013

Wyniki wyszukiwania

On modules and rings with the restricted minimum condition

An intermediate ring between a polynomial ring and a power series ring