On the existence of nonzero injective covers and projective envelopes of modules | Zendy

Xiaoxiang Zhang | Zendy; Song Xian-mei | Zendy

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On the existence of nonzero injective covers and projective envelopes of modules

Author(s) -

Xiaoxiang Zhang,

Song Xian-mei

Publication year - 2013

Publication title -

turkish journal of mathematics

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.454

H-Index - 27

eISSN - 1303-6149

pISSN - 1300-0098

DOI - 10.3906/mat-1203-24

Subject(s) - mathematics , injective function , cover (algebra) , projective cover , envelope (radar) , ring (chemistry) , injective module , simple module , primitive ring , simple (philosophy) , pure mathematics , zero (linguistics) , projective test , combinatorics , principal ideal ring , projective space , collineation , commutative ring , computer science , philosophy , chemistry , engineering , telecommunications , epistemology , commutative property , mechanical engineering , radar , organic chemistry , linguistics

In general, the injective cover (projective envelope) of a simple module can be zero. A ring R is called a weakly left V-ring (strongly left Kasch ring) if every simple left R-module has a nonzero injective cover (projective envelope). It is proven that every nonzero left R-module has a nonzero injective cover if and only if R is a left Artinian weakly left V-ring. Dually, every nonzero left R-module has a nonzero projective envelope if and only if R is a left perfect right coherent strongly left Kasch ring. Some related rings and examples are considered. © Tübi̇tak.

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