Structure of Rings
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a) Determine the structure of the ring R obtained from Z by adjoining an element w satisfying each set of relations: (i) 2w-6=0, w-10=0 and (ii) w3+w2+1=0, w2+w=0.
b) Let f=x4+x3+x2+x+1 and let y denote the residue of x in the ring R=Z[x]/(f). Express (y3+y2+y)(y5+1) in terms of the basis (1,y,y2,y3) of R.
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This solution helps find the structure of rings.
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a) (i) The ring R is isomorphic to Z[w]/(2w-6, w-10). But Z[w]/(w-10) is isomorphic to Z. The isomorphism is given by sending the residue of the polynomial f(w) to f(10). Under this isomorphism the residue of (2w-6) is mapped to ...
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