Embeddings of strongly right bounded rings and algebras.

*(English)*Zbl 0678.16024A ring (algebra) R is said to be strongly right bounded if every non-zero right ideal of R contains a non-zero ideal. In the paper, several sets of necessary and sufficient conditions for an algebra to be embeddable in a strongly right bounded algebra with a unity are found. The following result is fundamental in this respect: Let T be an R-algebra, R being a non-zero commutative ring with unity. Then the standard Dorroh extension S of T is a strongly right bounded R-algebra iff T is a strongly right bounded R-algebra satisfying the following condition: If \(x\in T\) and \(r\in R\) are such that \(r\neq 0\), \(xt+rt=0\) for each \(t\in T\) and either xT\(\neq 0\) or \(xT=0\), Tx\(\neq 0\) and \(r^ 2=0\), then there exists an ideal I of R such that either Ix\(\neq 0\) or Ir\(\neq 0\) and, for each \(i\in I\), there exists \(j\in I\) with \(ir=jr\) and \(i(tx+rt+x)=jx\) for all \(t\in T\). Beside this result and four other corollaries, the paper is supplied with several interesting examples.

Reviewer: T.Kepka

##### MSC:

16Dxx | Modules, bimodules and ideals in associative algebras |

16S20 | Centralizing and normalizing extensions |

16W99 | Associative rings and algebras with additional structure |

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\textit{G. F. Birkenmeier} and \textit{H. E. Heatherly}, Commun. Algebra 17, No. 3, 573--586 (1989; Zbl 0678.16024)

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