57 457
Assignments Done
Successfully Done
In February 2018
Your physics homework can be a real challenge, and the due date can be really close — feel free to use our assistance and get the desired result.
Be sure that math assignments completed by our experts will be error-free and done according to your instructions specified in the submitted order form.
Our experts will gladly share their knowledge and help you with programming homework. Keep up with the world’s newest programming trends.

Answer on Algebra Question for sanches

Question #16870
Show that, if A = R[x] where R is a commutative ring, then for f =(sum: i)a_ix^i ∈ A: f ∈ U(A) ⇐⇒ a0 ∈ U(R) and ai is nilpotent for i ≥ 1.
Expert's answer
The “if” part is clear since, in anycommutative ring, the sum of a unit and a nilpotent element is always a unit.
For the “only if” part, a rather slick proof using basic commutative algebra
(mainly the fact that the intersection of prime ideals in R is equal toNil(R)). For the sake of completeness, let us record below a proof thatuses a “bare-hands” approach, accessible to every beginning student in algebra.
Assume that f ∈ U(A). It is easy to see that a0∈ U(R). We are done if we can show that an is nilpotent in casen ≥ 1 (for then f − anxn is also a unit, andwe can repeat the argument). Let I = {b ∈ R : antb = 0 forsome t ≥ 1}.
Say fg = 1, where g = b0+ · · · + bmxm ∈ A. We claim that bi ∈ I for every i. If this is the case, there exists t≥ 1 such that antbi = 0 for all i,and so ant = f ant g =0, as desired. The claim is certainly true if i = m. Inductively,if bi+1, . . . , bm ∈ I, then, comparing the coefficients of xn+iin the equation fg =1, we have anbi + an−1bi+1+ an−2bi+2 + · · ·= 0. For s sufficiently large, we then have as+1nbi = −an−1asnbi+1−· · · = 0, so bi ∈ I as claimed.

Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!


No comments. Be first!

Leave a comment

Ask Your question