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# R ring for each nonzero a∈R, exist unique b∈R,aba=b,Show that R has no divisors of 0, bab=b, and has identity?

Let R be a ring that contains at least two elements. Suppose for each nonzero a∈R, there exists a unique b∈R such that aba=a.

Show that R has no divisors of zero.

Show that bab=b.

Show that R has identity.

Show that R is a division ring.

### 3 Answers

- 9 years agoFavorite Answer
Assume that R has a divisor of 0, say a different than 0, thus there exists c in R\{0} such that ac = 0, take b such that aba= a, thus ab = ab -ac = a(b-c), and from there, a= aba = a(b-c)a, by the uniqueness hypothesis we have that b = b -c , thus c=0, a contradiction.

If aba = a, then abab = ab thus abab -ab = 0, thus a( bab - b)=0, as R has no divisors of 0 we have that bab = b.

Take any a different than 0, and b such that aba = a. Let c be any element of R

aba = a implies abab = ab, thus abc = ababc thus ab( c - abc) = 0, as ab is not 0 and R has no divisors of 0 we obtain that c = abc.

aba = a implies abab = ab, thus cab = cabab thus (c - cab) ab = 0, as ab is not 0 and R has no divisors of 0 we obtain that c = cab. thus ab is an identity for R

We showed that if a is an element distinct of 0 there is a unique b such that ab = identity (the uniqueness of the hypothesis of b) using only that a and b satisfy aba=a; as bab= b we would obtain that ba = identity. Thus R is a division ring

Source(s): my brain - karpLv 44 years ago
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