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On rings with ascending chain conditions on right annihilators. (English) Zbl 1077.16023
Let $$R$$ be an associative ring with identity. The ring $$R$$ is called: (i) orthogonally finite if $$R$$ has no infinite sets of non-zero orthogonal idempotents; (ii) right Utumi if $$R$$ is right non-singular and every non-essential right ideal of $$R$$ has non-zero left annihilator; (iii) of bounded index (of nilpotence) if there exists a positive integer $$n$$ such that $$a^n=0$$ for each nilpotent element $$a\in R$$; (iv) right p-injective if for any principal right ideal $$I$$ of $$R$$, any homomorphism $$I\to R$$ extends to an endomorphism of $$R$$.
The author shows that $$R$$ has ACC on right annihilators if and only if $$R$$ is orthogonally finite and $$R$$ has ACC on right annihilators containing no non-zero idempotents. Also, the property that $$R$$ has ACC on right annihilators is characterized for several types of rings $$R$$, such as right non-singular right p-injective, right Utumi or semiprime of bounded index.
##### MSC:
 16P60 Chain conditions on annihilators and summands: Goldie-type conditions 16D80 Other classes of modules and ideals in associative algebras 16D50 Injective modules, self-injective associative rings