ely generated modules 1.1. Unless speci ed otherwise, in this section we will work with an arbitrary (not neces. arily commutative) ring A. By an A-module we. will mean a left A-module. …

Any finitely generated module is a quotient of Rn for some n, so it suffices to classify the submodules of Rn. The proof of the previous theorem shows that these are exactly the direct …

Math 103A – Modern algebra I Lecture 12: Finitely generated abelian groups Lucas Buzaglo Based on the textbook A first course in abstract algebra by Fraleigh and Brand November 4, …

This theorem describes in precise detail the structure of a finitely-generated module over a P.I.D. Recall that if R is any ring, then an R-module M is an abelian group (we’ll use + as the …

Every prime divisor of n must divide n1. If n is the product of distinct primes and G is an Abelian group of order n, then G = Zn. 1 : : : pak . Then, The decomposition given above is unique. …

e M is finitely generated. Then this is the same as saying that there is an exact equence A⊕n → M → 0. The image of your favorite set of generators in A⊕n give

Every field K has a minimal subfield, isomorphic to either Q or Fp for some prime p. Call K finitely generated (f.g.) if it is finitely generated as a field extension of its minimal subfield. F.g. fields …