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Embeddings of strongly right bounded rings and algebras. (English) Zbl 0678.16024
A 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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##### References:
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