Cancellation and stability properties of generalized torsion modules. (English) Zbl 1484.16002
Let \(R\) be a ring and \(M, M'\) be arbitrary \(R\)-modules. Recall that the cancellation problem for any \(R\)-module \(N\) asks, if \(M \oplus N \cong M' \oplus N\) implies \(M \cong M'\). The scope of this paper is a restriction of the cancellation problem to finitely generated \(R\)-modules and modification of the statement to finding all modules \(M'\) that satisfy for any positive integers \(a,b\), \(M \oplus R^a \cong M' \oplus R^b\) implies \(M \cong M'\). The isomorphism classes of such modules \(M'\) is called the stability class of \(M\) and denoted by \([M]\). This paper considers the tree structure on \([M]\) when \(M\) is a finitely generated module over a weakly finite ring \(R\). First, the paper studies the tree structure on \([0]\) and further proves that if \(M\) is also a generalized torsion module then the tree structure on \([0]\) and \([M]\) are isomorphic. Last section provides examples of group rings and application of the previous results to modules over these group rings and compares the results to the current literature.
Reviewer: Muge Kanuni (Düzce)
MSC:
| 16D70 | Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) |
| 16S90 | Torsion theories; radicals on module categories (associative algebraic aspects) |
| 20E06 | Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations |
References:
| [1] | V. A. Artamonov, “The quantum Serre problem”, Uspekhi Mat. Nauk 53:4(322) (1998), 3-76. In Russian. · Zbl 0931.16002 · doi:10.1070/rm1998v053n04ABEH000056 |
| [2] | H. Bass, “Projective modules over free groups are free”, J. Algebra 1 (1964), 367-373. · Zbl 0145.26604 · doi:10.1016/0021-8693(64)90016-X |
| [3] | P. H. Berridge and M. J. Dunwoody, “Nonfree projective modules for torsion-free groups”, J. London Math. Soc. (2) 19:3 (1979), 433-436. · Zbl 0399.20005 · doi:10.1112/jlms/s2-19.3.433 |
| [4] | P. M. Cohn, “Some remarks on the invariant basis property”, Topology 5 (1966), 215-228. · Zbl 0147.28802 · doi:10.1016/0040-9383(66)90006-1 |
| [5] | P. M. Cohn, Skew fields: Theory of general division rings, Encyclopedia of Mathematics and its Applications 57, Cambridge University Press, 1995. · Zbl 0840.16001 · doi:10.1017/CBO9781139087193 |
| [6] | H. Jacobinski, “Genera and decompositions of lattices over orders”, Acta Math. 121 (1968), 1-29. · Zbl 0167.04503 · doi:10.1007/BF02391907 |
| [7] | F. E. A. Johnson, “The stable class of the augmentation ideal”, K-Theory 34:2 (2005), 141-150. · Zbl 1094.20001 · doi:10.1007/s10977-005-3937-1 |
| [8] | F. E. A. Johnson, Syzygies and homotopy theory, Algebra and Applications 17, Springer, 2012. · Zbl 1233.13001 · doi:10.1007/978-1-4471-2294-4 |
| [9] | F. E. A. Johnson, “A cancellation theorem for generalized Swan modules”, Illinois J. Math. 63:1 (2019), 103-125. · Zbl 1447.16005 · doi:10.1215/00192082-7600042 |
| [10] | M. C. Kang, “Metacyclic algebras”, pp. 165-171 in Algebraic structures and number theory (Hong Kong, 1988), edited by S. P. Lam and K. P. Shum, World Sci. Publ., Teaneck, NJ, 1990. · Zbl 1439.12004 |
| [11] | L. Klingler, Modules over the integral group ring of a nonabelian group of order pq, Mem. Amer. Math. Soc. 341, Amer. Math. Soc., Providence, RI, 1986. · Zbl 0598.16014 · doi:10.1090/memo/0341 |
| [12] | S. Montgomery, “von Neumann finiteness of tensor products of algebras”, Comm. Algebra 11:6 (1983), 595-610. · Zbl 0488.16015 · doi:10.1080/00927878308822867 |
| [13] | D. Quillen, “Projective modules over polynomial rings”, Invent. Math. 36 (1976), 167-171. · Zbl 0337.13011 · doi:10.1007/BF01390008 |
| [14] | A. A. Suslin, “Projective modules over polynomial rings are free”, Dokl. Akad. Nauk SSSR 229:5 (1976), 1063-1066. In Russian. |
| [15] | R. G. Swan, K-theory of finite groups and orders, Lecture Notes in Mathematics 149, Springer, 1970. · Zbl 0205.32105 |
| [16] | R. G. Swan, “Projective modules over Laurent polynomial rings”, Trans. Amer. Math. Soc. 237 (1978), 111-120. · Zbl 0404.13006 · doi:10.2307/1997613 |
| [17] | R. G. Swan, “Projective modules over binary polyhedral groups”, J. Reine Angew. Math. 342 (1983), 66-172. · Zbl 0503.20001 · doi:10.1515/crll.1983.342.66 |
| [18] | R. G. Swan, “Torsion free cancellation over orders”, Illinois J. Math. 32:3 (1988), 329-360 · Zbl 0666.16002 |
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