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Assume R is a ring, M and N are r-modules. Show that any two r-m.pdf

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Assume R is a ring, M and N are r-modules. Show that any two r-modules with the same rank are isomorphic. Solution If M and N are R-modules of rank k, then there are bases{m1,m2,...,mk} and{n1,n2,...,nk} of M and N. Then, the homomorphism f from M to N, defined by f(mi) =f(ni) is an isomorphism. This shows that every twoR-modules of the same rank are isomorphic..

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Assume R is a ring, M and N are r-modules. Show that any two r-modules with the same rank are isomorphic. Solution If M and N are R-modules of rank k, then there are bases{m1,m2,...,mk} and{n1,n2,...,nk} of M and N. Then, the homomorphism f from M to N, defined by f(mi) =f(ni) is an isomorphism. This shows that every twoR-modules of the same rank are isomorphic.

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Assume R is a ring, M and N are r-modules. Show that any two r-m.pdf

  • 1. Assume R is a ring, M and N are r-modules. Show that any two r-modules with the same rank are isomorphic. Solution If M and N are R-modules of rank k, then there are bases{m1,m2,...,mk} and{n1,n2,...,nk} of M and N. Then, the homomorphism f from M to N, defined by f(mi) =f(ni) is an isomorphism. This shows that every twoR-modules of the same rank are isomorphic.