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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:
[1] Birkenmeier G. F., Algebra Univ.
[2] DOI: 10.1090/S0002-9904-1932-05333-2 · Zbl 0003.38701
[3] Faith C., London Math. Soc. Lecture Notes Series 88 (1984)
[4] DOI: 10.1090/S0002-9947-1958-0098763-0
[5] Jacobson N., Amer. Math. Soc. Math. Surveys 1 (1943)
[6] Jezek J., Czech. Math 34 pp 396– (1984)
[7] DOI: 10.1007/BF02572495 · Zbl 0525.20046
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