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Found problems: 2

2009 Romania National Olympiad, 2
[b]a)[/b] Show that the set of nilpotents of a finite, commutative ring, is closed under each of the operations of the ring. [b]b)[/b] Prove that the number of nilpotents of a finite, commutative ring, divides the number of divisors of zero of the ring.
nilpotent abstract algebra commutative algebra ring theory algebra number theory

2007 Nicolae Păun, 2
For a given natural number, $ n\ge 2, $ consider two matrices $ A,B\in\mathcal{M}_n(\mathbb{C}) $ that commute and such that $ A $ is invertible and that the function $ M:\mathbb{C}\longrightarrow\mathbb{C} ,M(x)=\det (A+xB) $ is bounded above or below. Prove that $ B^n=0. $ [i]Sorin Rădulescu[/i] and [i]Ion Savu[/i]
matrix function linear algebra commutative algebra algebra matrices