# On rings close to regular and $p$-injectivity

Commentationes Mathematicae Universitatis Carolinae (2006)

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

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topMing, 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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