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This article is cited in 1 scientific paper (total in 1 paper)
Orders in Uniserial Rings
A. A. Tuganbaev Moscow Power Engineering Institute (Technical University)
Abstract:
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.
Received: 28.07.2001
Citation:
A. A. Tuganbaev, “Orders in Uniserial Rings”, Mat. Zametki, 74:6 (2003), 924–933; Math. Notes, 74:6 (2003), 874–882
Linking options:
https://www.mathnet.ru/eng/mzm320https://doi.org/10.4213/mzm320 https://www.mathnet.ru/eng/mzm/v74/i6/p924
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Abstract page: | 388 | Full-text PDF : | 193 | References: | 52 | First page: | 4 |
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