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Pure Baer injective modules. (English) Zbl 0886.16003
Let $$R$$ be a ring with identity. All modules are unitary left $$R$$-modules and all homomorphisms are $$R$$-homomorphisms. A left $$R$$-module $$M$$ is called pure-split if every pure submodule of $$M$$ is a direct summand. $$R$$ is said to be left pure split if $$_RR$$ is pure-split. A non-zero $$R$$-module $$M$$ is called pure simple if $$\{0\}$$ and $$M$$ are its only submodules.
The author gives the following definitions: 1) An $$R$$-module $$M$$ is called a pure Baer injective module if for each pure left ideal $$I$$ of $$R$$, any $$R$$-homomorphism $$f\colon I\to M$$ can be extended to an $$R$$-homomorphism $$\overline f\colon R\to M$$; 2) A ring $$R$$ is called left pure hereditary if every pure left ideal of $$R$$ is projective; 3) A ring $$R$$ is called an SSBI-ring if every semisimple $$R$$-module is pure Baer injective; 4) A left $$R$$-module $$M$$ is called $$\Sigma$$-pure Baer injective if every direct sum of copies of $$M$$ is pure Baer injective.
Some of the main results established by the author are: 1) The following statements are equivalent: (i) $$R$$ is left pure hereditary. (ii) The homomorphic image of a pure Baer injective $$R$$-module is pure Baer injective. (iii) Any finite sum of injective submodules of an $$R$$-module is pure Baer injective; 2) Every direct sum of copies of $$R$$ is pure-split if and only if every flat $$R$$-module is pure-split; 3) Let $$M$$ be an $$R$$-module in which every cyclic submodule is pure-split, then every non-zero submodule of $$M$$ contains a pure-simple submodule; 4) A ring which is both a $$V$$-ring and an SSBI-ring satisfies the ascending chain condition on pure left ideals; 5) A ring $$R$$ in which every injective $$R$$-module is $$\Sigma$$-pure Baer injective, satisfies the ascending chain condition on pure left ideals.

##### MSC:
 16D50 Injective modules, self-injective associative rings 16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) 16P70 Chain conditions on other classes of submodules, ideals, subrings, etc.; coherence (associative rings and algebras)
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