An R-module is called cyclic if it is generated by a single element. (a) Show that an R-module is cy ...

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An R-module is called cyclic if it is generated by a single element. (a) Show that an R-module is cyclic if and only if it is isomorphic to R/I for some ideal I. (b) Show that the ideal I in part (a) is unique if R is commutative. (c) Let F be a field, and let M_n(F) be the ring of n x n matrices with coefficients in F. The vector space F^n, considered as a space of column vectors, is a left module over M_n(F) via matrix multiplication. Prove that it is a cyclic module, and find a left ideal I of M_n(F) such that M_n(F)/I is isomorphic to F^n. Is there more than one ideal with this property?