Extending module homomorphism to localized module

Suppose M is an A-module and N an S^-1A-module, f:M→N an A-linear map, τ:M→S^-1M the canonical map. Then there is a unique S^-1A-linear module homomorphism g:S^-1M→N such that f=g∗τ.

k[x](x) is not a polynomial ring

k[x]_(x) is a DVR, and as such, has a unique(up to associates) irreducible element. Polynomial ring over an integral domain, on the other hand, has at least two irreducible elements x and x+1 that are not associates.

A a UFD, S⊆A a saturated multiplicative subset not containing 0. Then S-1A is a UFD.

Lemma 1. S contains all the units in A.
Lemma 2. In general, if an element of a ring is irreducible then so are its associates.
Proposition 3. If a∈A\S is irreducible in A then it is irreducible in S^-1A.
Lemma 4. a∈A is a unit in S^-1A if and only if a∈S.
Proposition 5. If a is irreducible in S^-1A then it is nonzero, nonunit in A, and its factorization is of the form a=s₁s₂…s_rb, b∉S.
Proposition 6.S^-1A is a UFD.