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Transactions of the American Mathematical Society

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A noncommutative Hilbert basis theorem and subrings of matrices


Author: S. A. Amitsur
Journal: Trans. Amer. Math. Soc. 149 (1970), 133-142
MSC: Primary 16.25
DOI: https://doi.org/10.1090/S0002-9947-1970-0258869-5
MathSciNet review: 0258869
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Abstract: A finitely generated central extension $ A[{u_1}, \ldots ,{u_k}]$ of a commutative noetherian ring A, satisfies the ascending chain condition for ideals P for which $ A[{u_1}, \ldots ,{u_k}]/P$ can be embedded in matrix rings $ {M_n}(K)$ over arbitrary commutative rings K and n bounded. The method of proof leads to an example of a ring R which satisfies the same identities of $ {M_n}(K)$ but nevertheless cannot be embedded in any matrix ring over a commutative ring of arbitrary finite order.


References [Enhancements On Off] (What's this?)

  • [1] S.A. Amitsur and C. Procesi, Jacobson-rings and Hilrt algebras with polynomial identities, Ann. Mat. Pura Appl. (4) 71 (1966), 61-72. MR 34 #5869. MR 0206044 (34:5869)
  • [2] C. Procesi, Non-commutative affine rings, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. I (8) 8 (1967) 237-255. MR 37 #256. MR 0224657 (37:256)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1970-0258869-5
Keywords: Hilbbert basis theorem, generic matrices, ascending chain condition, finitely generated rings with identities
Article copyright: © Copyright 1970 American Mathematical Society

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