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Found problems: 1

2008 IMS, 5
Prove that there does not exist a ring with exactly 5 regular elements. ($ a$ is called a regular element if $ ax \equal{} 0$ or $ xa \equal{} 0$ implies $ x \equal{} 0$.) A ring is not necessarily commutative, does not necessarily contain unity element, or is not necessarily finite.
group theory regular element abstract algebra ring theory superior algebra Chinese remainder theorem