On rings close to regular and p -injectivity

Roger Yue Chi Ming

Commentationes Mathematicae Universitatis Carolinae (2006)

  • Volume: 47, Issue: 2, page 203-212
  • ISSN: 0010-2628

Abstract

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The following results are proved for a ring A : (1) If A is a fully right idempotent ring having a classical left quotient ring Q which is right quasi-duo, then Q is a strongly regular ring; (2) A has a classical left quotient ring Q which is a finite direct sum of division rings iff A is a left TC -ring having a reduced maximal right ideal and satisfying the maximum condition on left annihilators; (3) Let A have the following properties: (a) each maximal left ideal of A is either a two-sided ideal of A or an injective left A -module; (b) for every maximal left ideal M of A which is a two-sided ideal, A / M A is flat. Then, A is either strongly regular or left self-injective regular with non-zero socle; (4) A is strongly regular iff A is a semi-prime left or right quasi-duo ring such that for every essential left ideal L of A which is a two-sided ideal, A / L A is flat; (5) A prime ring containing a reduced minimal left ideal must be a division ring; (6) A commutative ring is quasi-Frobenius iff it is a YJ -injective ring with maximum condition on annihilators.

How to cite

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Ming, Roger Yue Chi. "On rings close to regular and $p$-injectivity." Commentationes Mathematicae Universitatis Carolinae 47.2 (2006): 203-212. <http://eudml.org/doc/249845>.

@article{Ming2006,
abstract = {The following results are proved for a ring $A$: (1) If $A$ is a fully right idempotent ring having a classical left quotient ring $Q$ which is right quasi-duo, then $Q$ is a strongly regular ring; (2) $A$ has a classical left quotient ring $Q$ which is a finite direct sum of division rings iff $A$ is a left $\operatorname\{TC\}$-ring having a reduced maximal right ideal and satisfying the maximum condition on left annihilators; (3) Let $A$ have the following properties: (a) each maximal left ideal of $A$ is either a two-sided ideal of $A$ or an injective left $A$-module; (b) for every maximal left ideal $M$ of $A$ which is a two-sided ideal, $A/M_A$ is flat. Then, $A$ is either strongly regular or left self-injective regular with non-zero socle; (4) $A$ is strongly regular iff $A$ is a semi-prime left or right quasi-duo ring such that for every essential left ideal $L$ of $A$ which is a two-sided ideal, $A/L_A$ is flat; (5) $A$ prime ring containing a reduced minimal left ideal must be a division ring; (6) A commutative ring is quasi-Frobenius iff it is a $\operatorname\{YJ\}$-injective ring with maximum condition on annihilators.},
author = {Ming, Roger Yue Chi},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {strongly regular; $p$-injective; $\operatorname\{YJ\}$-injective; biregular; von Neumann regular; strongly regular; -injective; YJ-injective; biregular; van Neumann regular},
language = {eng},
number = {2},
pages = {203-212},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On rings close to regular and $p$-injectivity},
url = {http://eudml.org/doc/249845},
volume = {47},
year = {2006},
}

TY - JOUR
AU - Ming, Roger Yue Chi
TI - On rings close to regular and $p$-injectivity
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2006
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 47
IS - 2
SP - 203
EP - 212
AB - The following results are proved for a ring $A$: (1) If $A$ is a fully right idempotent ring having a classical left quotient ring $Q$ which is right quasi-duo, then $Q$ is a strongly regular ring; (2) $A$ has a classical left quotient ring $Q$ which is a finite direct sum of division rings iff $A$ is a left $\operatorname{TC}$-ring having a reduced maximal right ideal and satisfying the maximum condition on left annihilators; (3) Let $A$ have the following properties: (a) each maximal left ideal of $A$ is either a two-sided ideal of $A$ or an injective left $A$-module; (b) for every maximal left ideal $M$ of $A$ which is a two-sided ideal, $A/M_A$ is flat. Then, $A$ is either strongly regular or left self-injective regular with non-zero socle; (4) $A$ is strongly regular iff $A$ is a semi-prime left or right quasi-duo ring such that for every essential left ideal $L$ of $A$ which is a two-sided ideal, $A/L_A$ is flat; (5) $A$ prime ring containing a reduced minimal left ideal must be a division ring; (6) A commutative ring is quasi-Frobenius iff it is a $\operatorname{YJ}$-injective ring with maximum condition on annihilators.
LA - eng
KW - strongly regular; $p$-injective; $\operatorname{YJ}$-injective; biregular; von Neumann regular; strongly regular; -injective; YJ-injective; biregular; van Neumann regular
UR - http://eudml.org/doc/249845
ER -

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