El Bashir, Robert; Kepka, Tomáš On when small semiprime rings are slender. (English) Zbl 0854.16015 Commun. Algebra 24, No. 5, 1575-1580 (1996). For non-commutative rings \(R\) of cardinality \(<2^{\aleph_0}\) that are either prime, right Noetherian semi-prime or strongly regular rings, it is shown that \(R\) is slender iff no non-zero (ring) direct summand of \(R\) is completely reducible. One result is as follows: If \(R\) is a small prime ring, then \(R\) is slender iff \(R\) is not isomorphic to a full matrix ring over a division ring.Reviewer’s remark: Cf. the reviewer’s following result from 1982 [Dissertation, Tulane Univ.]: An \(R\)-module \(M\) is slender iff \(\text{Hom}_R(\bigoplus^n_1R,M)\) is slender as an \(M_n(R)\)-module. Consequently, the ring \(R\) is slender iff the ring of \(n\times n\) matrices \(M_n(R)\) is slender. Reviewer: R.Dimitrić (Berkeley) Cited in 2 Documents MSC: 16N60 Prime and semiprime associative rings 16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) 16D80 Other classes of modules and ideals in associative algebras Keywords:slender rings; prime rings; semiprime rings; small modules; strongly regular rings; direct summands; small prime rings; rings of matrices × Cite Format Result Cite Review PDF Full Text: DOI References: [1] A Uouch D., Modules maigres · Zbl 0222.13010 [2] Anderson F. W., Rings and Categories of Modules (1992) · Zbl 0765.16001 [3] DOI: 10.1080/00927878308822927 · Zbl 0578.13010 · doi:10.1080/00927878308822927 [4] DOI: 10.1007/BFb0103717 · doi:10.1007/BFb0103717 [5] Eda K., Tsukuba J. Math 6 pp 187– (1982) [6] DOI: 10.1016/0021-8693(83)90174-6 · Zbl 0538.20027 · doi:10.1016/0021-8693(83)90174-6 [7] Eda K., Fundamenta Math 135 pp 5– (1990) [8] Ehrenfeueht A., Bull. Acad. Polon. Sci., Sér. Sci. Math 2 pp 261– (1954) [9] Eklof, P. and Mekler, A. 1990. ”Almost Free Modules”. New York, North-Holland · Zbl 0718.20027 [10] Fuchs, L. 1960. ”Abelian Groups”. Pergamon Press. · Zbl 0100.02803 [11] Fuchs, L. 1970. ”Infinite Abelian Groups”. Vol. I, Academic Press. · Zbl 0209.05503 [12] Fuchs, L. 1973. ”Infinite Abelian Groups”. Vol. II, Academic Press. · Zbl 0257.20035 [13] Fuchs, L. and Sake, L. 1985. ”Modules over Valuation Domains”. Marcel Dekker. [14] Heinlein, G. 1971. ”Vollreflexive Ringe und schlanke Moduln”. Erlangen: Dissertation. [15] Lady E., Pacific J. Math 49 pp 397– (1973) · Zbl 0274.16015 · doi:10.2140/pjm.1973.49.397 [16] Mader A., in: Abelian Groups and Modules, CISM Courses and Lectures 49 pp 315– (1984) · doi:10.1007/978-3-7091-2814-5_23 [17] De Marco G., Symposia Math 13 pp 153– (1974) [18] Nunkee R., Acta Sci. Math. Szeged 23 pp 67– (1962) [19] Salce L., Ann. Univ. Ferrara, Ser. VH-Sc. Mat 20 pp 59– (1975) [20] Sasiada E., Bull. Acad. Polon. Sci 7 pp 143– (1959) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.