# Ideals : Cyclic Module in a Commutative Ring

Prove that every cyclic module in a commutative ring R is of the form R/L for some ideal L. //Note: R/L is said "R modulo L."

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Prove that every cyclic module in a commutative ring R is of the form R/L for some ideal L. //Note: R/L is said "R modulo L"

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Definitions (from Fraleigh)

-- Let R be a ring. A (left) R-module consists of an abelian group M together with an operation of external multiplication of each element of M by each element of R on the left such that for all a, b elements of M and r,s elements of R,

(1) ra is an element of M, (2) r(a + b) = ra + rb, (3) (r + s)a = ra + sa, (4) (rs)a = r(sa)

(It is *not* necessary for M to be a subset of R, but in the ...

#### Solution Summary

Ideals and a cyclic module in a commutative ring are investigated and discussed in the solution. The solution is detailed and well presented.