Geodesic
Almost completely decomposable groups with primary regulator quotients and their endomorphism rings
Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 2, pp. 17-38 
E. A. Blagoveshchenskaya. Almost completely decomposable groups with primary regulator quotients and their endomorphism rings. Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 2, pp. 17-38. http://geodesic.mathdoc.fr/item/FPM_2006_12_2_a1/
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     title = {Almost completely decomposable groups with primary regulator quotients and their endomorphism rings},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2006_12_2_a1/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let $X$ be a block-rigid almost completely decomposable group of ring type with regulator $A$ and $p$-primary regulator quotient $X/A$ such that $p^l=\exp X/A$ with natural $l>1$. From the well-known fact $p^l\operatorname{End}A\subset\operatorname{End}X\subset\operatorname{End}A$ it follows that $\operatorname{End}X=\operatorname{End}X\cap\operatorname{End}A$ and $p^l\operatorname{End}A=\operatorname{End}X\cap p^l\operatorname{End}A$. Generalizing these, we determine the chain $\operatorname{End}X=\mathcal E_A^{(l)}\subset\mathcal E_A^{(l-1)}\subset\mathcal E_A^{(l-2)}\subset\dots\subset\mathcal E_A^{(1)}\subset\mathcal E_A^{(0)}=\operatorname{End}A$, satisfying $p^{l-k}\mathcal E_A^{({k})}=\operatorname{End}X\cap p^{l-k}\operatorname{End}A$, and construct groups $X'_k$ and $\widetilde{X_k}$ such that $\mathcal E_A^{({k})}=\operatorname{Hom}(X'_k,\widetilde{X_k})$, where $k=1,2,\dots,l-1$.

[1] Blagoveschenskaya E. A., “O pryamykh razlozheniyakh abelevykh grupp bez krucheniya konechnogo ranga”, Zap. nauch. sem. LOMI AN SSSR, 132, 1983, 17–25 | Zbl

[2] Blagoveschenskaya E. A., “Razlozheniyakh abelevykh grupp konechnogo ranga bez krucheniya v pryamye summy nerazlozhimykh grupp”, Algebra i analiz, 4:2 (1992), 62–69 | MR

[3] Blagoveschenskaya E. A., “Avtomorfizmy kolets endomorfizmov blochno-zhëstkikh pochti vpolne razlozhimykh grupp”, Fundam. i prikl. mat., 10:2 (2004), 23–50 | Zbl

[4] Blagoveschenskaya E. A., Yakovlev A. V., “Pryamye razlozheniyakh abelevykh grupp konechnogo ranga bez krucheniya”, Algebra i analiz, 1:1 (1989), 111–127 | Zbl

[5] Kash F., Moduli i koltsa, Mir, M., 1981 | MR

[6] Krylov P. A., Mikhalëv A. V., Tuganbaev A. A., Svyazi abelevykh grupp i ikh kolets endomorfizmov, Tomsk, 2002

[7] Kurosh A. G., Teoriya grupp, Nauka, M., 1967 | MR | Zbl

[8] Lambek I., Koltsa i moduli, Mir, M., 1971 | MR | Zbl

[9] Fuks L., Beskonechnye abelevy gruppy, t. 1, Mir, M., 1974; т. 2, Мир, М., 1977

[10] Yakovlev A. V., “O pryamykh razlozheniyakh abelevykh grupp konechnogo ranga bez krucheniya”, Zap. nauch. sem. LOMI AN SSSR, 160, 1987, 272–285

[11] Arnold D., “Finite Rank Torsion Free Abelian Groups and Rings”, Lect. Notes Math., 931, Springer, 1982 | MR | Zbl

[12] Blagoveshchenskaya E., “Direct decompositions of almost completely decomposable Abelian groups”, Proc. Int. Conf. at Colorado Springs (CO, USA, August 7–12, 1995), Lect. Notes Pure Appl. Math., 182, ed. D. M. Arnold et al. Abelian Groups and Modules, New York: Marcel Dekker, 1996, 163–179 | MR | Zbl

[13] Blagoveshchenskaya E., “Classification of a class of almost completely decomposable groups”, Proc. Algebra Conf., Venezia 2002, Venice, Italy, June 3–8, Lect. Notes Pure Appl. Math., 236), ed. A. Facchini et al. Rings, Modules, Algebras, and Abelian Groups, New York: Marcel Dekker, 2002, 45–54 | MR

[14] Blagoveshchenskaya E., “Dualities between almost completely decomposable groups and their endomorphism rings”, J. Math. Sci., 131:5 (2005), 5948–5961 | DOI | MR | Zbl

[15] Blagoveshchenskaya E., Ivanov G., Schultz P., “The Baer–Kaplansky theorem for almost completely decomposable groups”, Contemp. Math., 273 (2001), 85–93 | MR | Zbl

[16] Blagoveshchenskaya E., Mader A., “Decompositions of almost completely decomposable abelian groups”, Contemp. Math., 171 (1994), 21–36 | MR | Zbl

[17] Corner A. L. S., “A note on rank and decomposition of torsion-free Abelian groups”, Math. Proc. Cambridge Philos. Soc., 57, 1961, 230–233 ; 1969; 66 ; 239–240 | MR | Zbl | MR | Zbl

[18] Faticoni T., Schultz P., “Direct decompositions of ACD groups with primary regulating index”, Abelian Groups and Modules (Proc. Int. Conf. at Colorado Springs, CO, USA, August 7–12, 1995), Lect. Notes Pure Appl. Math., 182, ed. D. M. Arnold et al., New York: Marcel Dekker, 1996, 233–241 | MR | Zbl

[19] Mader A., Almost Completely Decomposable Abelian Groups, Algebra, Logic and Applications, V. 13, Amsterdam: Gordon and Breach, 1999

[20] Mader A., Schultz P., “Endomorphism rings and automorphism groups of almost completely decomposable groups”, Comm. Algebra, 28 (2000), 51–68 | DOI | MR | Zbl

[21] Reid J., “Some matrix rings associated with ACD groups”, Abelian Groups and Modules, Proc. Int. Conf. (Dublin, Ireland, August 10–14), Trends in Mathematics, ed. C. Eklof et al., Birkhäuser, Basel, 1998, 191–198 | MR