Orders in Uniserial Rings

A. A. Tuganbaev

Matematičeskie zametki, Tome 74 (2003) no. 6, pp. 924-933.

Résumé

Let $A$ be a ring, and let $T(A)$ and $N(A)$ be the set of all the regular elements of $A$ and the set of all nonregular elements of $A$, respectively. It is proved that $A$ is a right order in a right uniserial ring if and only if the set of all regular elements of $A$ is a left ideal in the multiplicative semigroup $A$ and for any two elements $a_1$ and $a_2$ of $A$, either there exist two elements $b_1\in A$ and $t_1\in T(A)$ with $a_1b_1 = a_2t_1$ or there exist two elements $b_2\in A$ and $t_2\in T(A)$ with $a_2b_2 = a_1t_2$. A right distributive ring $A$ is a right order in a right uniserial ring if and only if the set $N(A)$ is a left ideal of $A$. If $A$ is a right distributive ring such that all its right divisors of zero are contained in the Jacobson radical $J(A)$ of $A$, then $A$ is a right order in a right uniserial ring.

Bibliographie

[1] Dubrovin N. I., The Rational Closure of Group Rings of Leftordered Groups, Gerhard Mercator Universität Duisburg Gesamthochschule, 1994

[2] Bessenrodt K., Brungs H. H., Törner G., Right Chain Rings, Part 1, Schriftenreihe des Fachbereich Mathematik, Universität Duisburg, 1990

[3] Bessenrodt K., Brungs H. H., Törner G., Right Chain Rings, Parts 2a; Parts 2b, Schriftenreihe des Fachbereich Mathematik, Universität Duisburg, 1992

[4] Dubrovin N. I., “Tsepnye oblasti”, Vestn. MGU. Ser. matem., mekh., 1980, no. 1, 51–54 | MR | Zbl

[5] Dubrovin N. I., “O tsepnykh koltsakh”, UMN, 37:4 (1982), 139–140 | MR | Zbl

[6] Behrens E. A., Ring Theory, Academic Press, New York, 1972

[7] Kon P., Svobodnye koltsa i ikh svyazi, Mir, M., 1975

[8] Tuganbaev A. A., Semidistributive Modules and Rings, Kluwer Academic Publishers, Dordrecht–Boston–London, 1998 | Zbl

[9] Tuganbaev A. A., Distributive Modules and Related Topics, Gordon and Breach, Amsterdam, 1999 | Zbl

[10] Stephenson W., “Modules whose lattice of submodules is distributive”, Proc. London Math. Soc., 28:2 (1974), 291–310 | DOI | MR | Zbl

[11] Menzel W., “Über den Untergruppenverband einer Abelschen Operatorgruppe. Teil II: Distributive und M-Verbande von Untergruppen einer Abelschen Operatorgruppe”, Math. Z., 74:1 (1960), 52–65 | DOI | MR | Zbl