MathDB

Problem 1

Part of 2006 VJIMC

Problems(2)

inequality over (0,1), with x_(i+1)-x_i<=h

Source: VJIMC 2006 1.1

6/28/2021
Given real numbers 0=x1<x2<…<x2n<x2n+1=10=x_1<x_2<\ldots<x_{2n}<x_{2n+1}=1 such that xi+1−xi≤hx_{i+1}-x_i\le h for 1≤i≤2n1\le i\le2n, show that 1−h2<∑i=1nx2i(x2i+1−x2i−1)<1+h2.\frac{1-h}2<\sum_{i=1}^nx_{2i}(x_{2i+1}-x_{2i-1})<\frac{1+h}2.
inequalities
when is u+v nilpotent for nilpotent u,v

Source: VJIMC 2006 2.1

6/28/2021
(a) Let uu and vv be two nilpotent elements in a commutative ring (with or without unity). Prove that u+vu+v is also nilpotent. (b) Show an example of a (non-commutative) ring RR and nilpotent elements u,v∈Ru,v\in R such that u+vu+v is not nilpotent.
Ring Theoryabstract algebra