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  • ...{\times}</math> is abelian, we can use the Fundamental Theorem of Finitely Generated Abelian Groups (FToFGAG) and we can write <math>k^{\times}</math> as a prod
    16 KB (2,660 words) - 22:42, 28 August 2024
  • ...{-1} = R(y_1/r)+\cdots+R(y_m/r)</math>, so <math>I^{-1}</math> is finitely generated as well. Now take <math>I^{-1} = Rx_1+\cdots +Rx_m</math>, and let <math>M_
    9 KB (1,648 words) - 15:36, 14 October 2017
  • # Every submodule <math>N</math> of <math>M</math> is [[finitely generated]] (i.e. can be written as <math>Rm_1+\cdots+Rm_k</math> for some <math>m_1, is a submodule of <math>M</math>, so it must be finitely generated, say
    3 KB (514 words) - 21:18, 10 April 2009
  • ...ational points on the curve, from which all further rational points may be generated.
    7 KB (1,102 words) - 16:23, 6 September 2008
  • A '''nilpotent group''' can be thought of a group that is only finitely removed from an [[abelian group]]. Specifically, it is a group <math>G</ma ...is in its image, for every prime <math>p</math>. Now, the order subgroup generated by the <math>P_p</math> must be divisible by every power of a prime that di
    9 KB (1,768 words) - 16:55, 5 June 2008
  • ...<math>X</math> containing <math>\mathcal{Q}</math>. This is the topology generated by <math>\mathcal{Q}</math>. Then the topology generated by <math>\mathcal{Q}</math> is the set containing <math>X</math>, and the s
    6 KB (1,142 words) - 14:38, 21 June 2008
  • ...>P \in A[[X]]</math>, let <math>I(P)</math> denote the two-sided [[ideal]] generated by the coefficients of <math>P</math>. The more general result states that ...\mathbb{Z}</math>, and <math>P</math> and <math>Q</math> power series with finitely many nonzero coefficients.
    3 KB (488 words) - 19:37, 28 September 2024
  • variables, then the [[ideal]] generated by these variables is not finitely generated.
    4 KB (617 words) - 18:59, 23 April 2023