Matrix wreath products of algebras and embedding theorems. (English) Zbl 1478.16018
Summary: We introduce a new construction of matrix wreath products of algebras that is similar to wreath products of groups. We then use it to prove embedding theorems for Jacobson radical, nil, and primitive algebras. In §6, we construct finitely generated nil algebras of arbitrary Gelfand-Kirillov dimension \(\geq 8\) over a countable field which answers a question from [J. P. Bell et al., Contemp. Math. 562, 41–52 (2012; Zbl 1267.16022)].
MSC:
| 16S99 | Associative rings and algebras arising under various constructions |
| 16N20 | Jacobson radical, quasimultiplication |
| 16N40 | Nil and nilpotent radicals, sets, ideals, associative rings |
| 16P90 | Growth rate, Gelfand-Kirillov dimension |
| 16S15 | Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting) |
Citations:
Zbl 1267.16022References:
| [1] | Alahmadi, Adel; Alsulami, Hamed, Wreath products by a Leavitt path algebra and affinizations, Internat. J. Algebra Comput., 24, 5, 707-714 (2014) · Zbl 1306.16002 · doi:10.1142/S0218196714500295 |
| [2] | Amitsur, A. S., Algebras over infinite fields, Proc. Amer. Math. Soc., 7, 35-48 (1956) · Zbl 0070.03004 · doi:10.2307/2033240 |
| [3] | Bartholdi, Laurent, Self-similar Lie algebras, J. Eur. Math. Soc. (JEMS), 17, 12, 3113-3151 (2015) · Zbl 1350.17005 · doi:10.4171/JEMS/581 |
| [4] | Bartholdi, Laurent; Erschler, Anna, Imbeddings into groups of intermediate growth, Groups Geom. Dyn., 8, 3, 605-620 (2014) · Zbl 1335.20046 · doi:10.4171/GGD/241 |
| [5] | Be\u\i dar, K. I., Radicals of finitely generated algebras, Uspekhi Mat. Nauk, 36, 6(222), 203-204 (1981) · Zbl 0482.16008 |
| [6] | Bell, Jason P., Examples in finite Gel’fand-Kirillov dimension, J. Algebra, 263, 1, 159-175 (2003) · Zbl 1028.16012 · doi:10.1016/S0021-8693(03)00021-8 |
| [7] | Bell, Jason P.; Small, Lance W., A question of Kaplansky, J. Algebra, 258, 1, 386-388 (2002) · Zbl 1020.16014 · doi:10.1016/S0021-8693(02)00513-6 |
| [8] | Bell, Jason P.; Small, Lance W.; Smoktunowicz, Agata, Primitive algebraic algebras of polynomially bounded growth. New trends in noncommutative algebra, Contemp. Math. 562, 41-52 (2012), Amer. Math. Soc., Providence, RI · Zbl 1267.16022 · doi:10.1090/conm/562/11129 |
| [9] | Bokut\cprime, L. A., Imbeddings into simple associative algebras, Algebra i Logika, 15, 2, 117-142, 245 (1976) · Zbl 0349.16007 |
| [10] | Borho, Walter; Kraft, Hanspeter, \"Uber die Gelfand-Kirillov-Dimension, Math. Ann., 220, 1, 1-24 (1976) · Zbl 0306.17005 · doi:10.1007/BF01354525 |
| [11] | Golod, E. S., On nil-algebras and finitely approximable \(p\)-groups, Izv. Akad. Nauk SSSR Ser. Mat., 28, 273-276 (1964) |
| [12] | Golod, E. S.; \v Safarevi\v c, I. R., On the class field tower, Izv. Akad. Nauk SSSR Ser. Mat., 28, 261-272 (1964) · Zbl 0136.02602 |
| [13] | Higman, Graham; Neumann, B. H.; Neumann, Hanna, Embedding theorems for groups, J. London Math. Soc., 24, 247-254 (1949) · Zbl 0034.30101 · doi:10.1112/jlms/s1-24.4.247 |
| [14] | Jacobson, Nathan, Structure of rings, American Mathematical Society Colloquium Publications, Vol. 37. Revised edition, ix+299 pp. (1964), American Mathematical Society, Providence, R.I. |
| [15] | Kaloujnine, L\'eo; Krasner, Marc, Le produit complet des groupes de permutations et le probl\`“eme d”extension des groupes, C. R. Acad. Sci. Paris, 227, 806-808 (1948) · Zbl 0038.16203 |
| [16] | Kaplansky, Irving, “Problems in the theory of rings” revisited, Amer. Math. Monthly, 77, 445-454 (1970) · Zbl 0208.29701 · doi:10.2307/2317376 |
| [17] | K\`“othe, Gottfried, Die Struktur der Ringe, deren Restklassenring nach dem Radikal vollst\'”andig reduzibel ist, Math. Z., 32, 1, 161-186 (1930) · JFM 56.0143.01 · doi:10.1007/BF01194626 |
| [18] | Krempa, Jan, Logical connections between some open problems concerning nil rings, Fund. Math., 76, 2, 121-130 (1972) · Zbl 0236.16004 · doi:10.4064/fm-76-2-121-130 |
| [19] | Lenagan, T. H.; Smoktunowicz, Agata, An infinite dimensional affine nil algebra with finite Gelfand-Kirillov dimension, J. Amer. Math. Soc., 20, 4, 989-1001 (2007) · Zbl 1127.16017 · doi:10.1090/S0894-0347-07-00565-6 |
| [20] | Lenagan, T. H.; Smoktunowicz, Agata; Young, Alexander A., Nil algebras with restricted growth, Proc. Edinb. Math. Soc. (2), 55, 2, 461-475 (2012) · Zbl 1259.16022 · doi:10.1017/S0013091510001100 |
| [21] | Mal\cprime cev, A. I., On a representation of nonassociative rings, Uspehi Matem. Nauk (N.S.), 7, 1(47), 181-185 (1952) · Zbl 0048.26001 |
| [22] | Markov, V. T., Matrix algebras with two generators and the embedding of PI-algebras, Uspekhi Mat. Nauk. Russian Math. Surveys, 47 47, 4, 216-217 (1992) · Zbl 0787.17003 · doi:10.1070/RM1992v047n04ABEH000926 |
| [23] | Neumann, B. H.; Neumann, Hanna, Embedding theorems for groups, J. London Math. Soc., 34, 465-479 (1959) · Zbl 0102.26401 · doi:10.1112/jlms/s1-34.4.465 |
| [24] | Olshanskii, Alexander Yu.; Osin, Denis V., A quasi-isometric embedding theorem for groups, Duke Math. J., 162, 9, 1621-1648 (2013) · Zbl 1331.20054 · doi:10.1215/00127094-2266251 |
| [25] | Petrogradsky, V. M.; Razmyslov, Yu. P.; Shishkin, E. O., Wreath products and Kaluzhnin-Krasner embedding for Lie algebras, Proc. Amer. Math. Soc., 135, 3, 625-636 (2007) · Zbl 1173.17008 · doi:10.1090/S0002-9939-06-08502-9 |
| [26] | Phillips, Richard E., Embedding methods for periodic groups, Proc. London Math. Soc. (3), 35, 2, 238-256 (1977) · Zbl 0365.20041 · doi:10.1112/plms/s3-35.2.238 |
| [27] | Smel\cprime kin, A. L., Wreath products of Lie algebras, and their application in group theory, Trudy Moskov. Mat. Ob\v{s}\v{c}., 29, 247-260 (1973) · Zbl 0286.17012 |
| [28] | Smoktunowicz, Agata; Bartholdi, Laurent, Jacobson radical non-nil algebras of Gel’fand-Kirillov dimension 2, Israel J. Math., 194, 2, 597-608 (2013) · Zbl 1283.16020 · doi:10.1007/s11856-012-0073-5 |
| [29] | Wilson, John S., Embedding theorems for residually finite groups, Math. Z., 174, 2, 149-157 (1980) · Zbl 0424.20028 · doi:10.1007/BF01293535 |
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