# Download e-book for kindle: Serre's Problem on Projective Modules by T.Y. Lam

By T.Y. Lam

ISBN-10: 3540233172

ISBN-13: 9783540233176

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Extra info for Serre's Problem on Projective Modules

Example text

12. Let R be a commutative ring, and let P = P (b1 , . . , bn ), where (b1 , . . , bn ) ∈ Umn (R). (1) P has a unimodular element iff b1 a1 + · · · + bn an = 0 for some (a1 , . . , an ) ∈ Umn (R). e. n is even), then P has a unimodular element. e. n = 3), then P has a unimodular element iff (b1 , b2 , b3 ) is completable. 4. Stably Free Modules and Hermite Rings 33 Proof. (1) We have P = ker(ϕ), where ϕ : R n → R is deﬁned by the row matrix (b1 , . . , bn ). Suppose the ai ’s exist. Then the column vector α := (a1 , .

In fact, if m ∈ M localizes to 0 in Mf and Mg , then f r m = 0 = g r m for some sufﬁciently large r. But f r and g r are still comaximal, so m = 0 ∈ M. For the rest, write S ={fn : n 0 }, T = { gn : n 0 }. Suppose ms ∈ Mf = MS and nt ∈ Mg = MT localize to the same element in Mfg . We may assume that sn = tm ∈ M. (In fact, we have (s t )(tm − sn) = 0 for some s ∈ S, t ∈ T . ) Write xs + yt = 1 (x, y ∈ R), and set q = xm + yn ∈ M. Then sq = (xs)m + y(sn) = (xs)m + (yt)m = m, and similarly, tq = n.

A commutative ring R is semilocal iff R/rad R is artinian. Proof. If R/rad R is artinian, it is a ﬁnite direct product of ﬁelds. Thus, Max R = Max (R/rad R) is ﬁnite. Conversely, if Max R = {m1 , . . , mr }, we have an embedding of R-modules R/rad R → ri=1 R/mi . Since the RHS has a composition series, the same is true for R/rad R. Thus, R/rad R satisﬁes both the ACC and the DCC for its ideals. 9. (1) Any commutative artinian ring is semilocal. g. as an R-module, then A is also semilocal. Proof.