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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.
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