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Let $(R,+,*)$ be an unitarian ring.

(a)

If $A$ is an unitarian ring.

$a,b$ are from $R$ and $a*b$ is invertible, it results that $a$ is invertible and $b$ is invertible?

(b) If $a$ is from $R$ and $a^n=a * a*...* a$ ($n$ or) is invertible,it results that $a$ is invertible?

(c) If $a$ is invertible to left and it isn't zero divisor to right,prove that $a$ is invertible.

(a) Here I tried to find a counterexample to that affirmation, but I couldn't find it.

(b) I wrote the definition of invertibility for a^n,but from that I couldn't find anything.

(c) I wrote the definition of invertibility and for non-zero divisibility for $a$, but I couldn't deduce anything from it.

Please, could you help me?

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Answer on (a):Not in general. See the answers to this question.

Answer on (b):

If $ba^n=1=a^nb$ then $(ba^{n-1})a=1=a(a^{n-1}b)$ showing that $a$ has a left- and a rightinverse. Consequently $a$ is invertible.

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  • $\begingroup$ Thank you very much for your help! $\endgroup$ – Adrian Berteanu Dec 12 '15 at 12:10
  • $\begingroup$ You are very welcome! $\endgroup$ – drhab Dec 12 '15 at 12:11

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