MathDB
Interesting condition on two elements of a certain ring

Source: Science ON 2021 grade XII/2

March 16, 2021
Ringsabstract algebrasuperior algebra

Problem Statement

Consider an odd prime pp. A comutative ring (A,+,⋅)(A,+, \cdot) has the property that ab=0ab=0 implies ap=0a^p=0 or bp=0b^p=0. Moreover, 1+1+⋯+1⏟p times=0\underbrace{1+1+\cdots +1}_{p \textnormal{ times}} =0. Take x,y∈Ax,y\in A such that there exist m,n≥1m,n\geq 1, m≠nm\neq n with x+y=xmy=xnyx+y=x^my=x^ny, and also yy is not invertible. \\ \\ <spanclass=′latex−bold′>(a)</span><span class='latex-bold'>(a)</span> Prove that (a+b)p=ap+bp(a+b)^p=a^p+b^p and (a+b)p2=ap2+bp2(a+b)^{p^2}=a^{p^2}+b^{p^2} for all a,b∈Aa,b\in A.\\ <spanclass=′latex−bold′>(b)</span><span class='latex-bold'>(b)</span> Prove that xx and yy are nilpotent.\\ <spanclass=′latex−bold′>(c)</span><span class='latex-bold'>(c)</span> If yy is invertible, does the conclusion that xx is nilpotent stand true? \\ \\ (Bogdan Blaga)
Back to ProblemsView on AoPS