1. Let R be a ring that contains at least two elements. Suppose for each nonzero a € R, there exists a unique be R such that aba - a. (i) Show that R has no divisors of zero. (ii) Show that bab = b. (iii) Show that R has a unity. (iv) Show that R is a division ring. 2. Let R be the commutative ring of all matrices with rational entries, M₁(Q), of the form abc 0 ab 00 a Show that the mapping : R Q given by - is a homomorphism. 4 ab c 0 ab 00 a ma

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