2706. RINGS6.1.11.Show that the setR.x/of rational functionsp.x/=q.x/, wherep.x/; q.x/2ROExŁandq.x/¤0, is a field. (Note the use of parenthesesto distinguish this ringR.x/of rational functions from the ringROExŁofpolynomials.)6.1.12.LetRbe a ring andXa set.Show that the set Fun.X; R/offunctions onXwith values inRis a ring. Show thatRis isomorphic to thesubring of constant functions onX. Show that Fun.X; R/is commutativeif, and only if,Ris commutative. Suppose thatRhas an identity; showthat Fun.X; R/has an identity and describe the units of Fun.X; R/.6.1.13.LetSEndK.V /, whereVis a vector space over a fieldK. ShowthatS0D fT2EndK.V /WTSDSTfor allS2Sgis a subring of EndK.V /.6.1.14.LetVbe a vector space over a fieldK. LetGbe a subgroup ofGL.V /. Show that the subring of EndK.V /generated byGis the set ofall linear combinationsPgnggof elements ofG, with coefficients inZ.
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