53 124
Assignments Done
97,7%
Successfully Done
In October 2017
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 Abstract Algebra Question for sanches

Question #17359
Let Ai (i ∈ I) be ideals in a ring R, and let A =(intersection on i) Ai. True or False: “If each R/Ai is von Neumann regular, then so is R/A”?
Expert's answer
If the indexing set I is infinite, the answer is “no”. For instance, taking Ap = (p)for primes p in R = Z, we have A =(intersection) Ap= (0). Here, each R/Ap ∼ Z/pZ is a field and hence von Neumann regular, but R/A ∼ Z is not von Neumann regular. To treat the case |I| < ∞,let I = {1, 2, . . . , n}. We claim that here the answer is “yes”. It suffices to prove this for n = 2, and we may assume A1 ∩ A2 = (0). Consider any a ∈ R. Since R/A1 and R/A2 are von Neumann regular, there exist x, y ∈ R such that (1 − ax)a ∈A1 and a(1 − ya) ∈A2. Then
(1 − ax)a(1 − ya)∈A1 ∩ A2 = 0.
This yields a = a(x+ y − xay)a, so R is von Neumann regular.

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!

Comments

No comments. Be first!

Leave a comment

Ask Your question