Math 312Assignment #3Due on Friday 30 October 2015•Except for “short-answer” questions, answers should be complete, clear, and neatly presentedto get full marks.•Only apropersubset of the following questions will be marked0. What did you do to celebrate Galois’s 204thbirthday on 25 October?1. She is called the mother of modern ring theory. A certain class of rings is named after her.Who is she? What class? Write down the definition.2. Identify all units inZn, inZ[i], inZ[x], inR[x].Short answers.3. Prove that{a2bb a:a, b∈Z}is a ring under the usual operations and is isomorphic toZ[√2] :={m+n√2 :m, n∈Z}.4. LetRbe a finite commutative ring with unity. Prove that every elementaofRis either a unitor a zero divisor.Hint:Consider the product ofawith members ofRand use the pigeon-hole principle.5.(a) LetRbe a ring anda∈R. Prove that each of the following is a left ideal(i) Ra,(ii){r∈R:ra= 0}.(b) What are the ideals in part (a) for the ringM2(Z) anda= [1 00 0]?6. LetR

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