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nilpotent+unit = unit; sufficient conditions to determine the nilpotent elements

Source: Romanian District Olympiad 2011, Grade XII, Problem 4

October 8, 2018

group theoryabstract algebraRing Theorynilpotencecardinalsuperior algebra

Problem Statement

Let be a ring

A.

Denote with

N(A)

the subset of all nilpotent elements of

A,

with

Z(A)

the center of

A,

and with

U(A)

the units of

A.

Prove:
a)

Z(A)=A\implies N(A)+U(A)=U(A) .

\text{card} (A)\in\mathbb{N}\wedge a+U(A)\subset U(A)\implies a\in N(A) .