270 6. RINGS 6.1.11. Show that the set R .x/ of rational functions p.x/=q.x/ , where p.x/;q.x/ 2 R ŒxŁ and q.x/ ¤0 , is a ﬁeld. (Note the use of parentheses to distinguish this ring R .x/ of rational functions from the ring R ŒxŁ of polynomials.) 6.1.12. Let R be a ring and X a set. Show that the set Fun .X;R/ of functions on X with values in R is a ring. Show that R is isomorphic to the subring of constant functions on X . Show that Fun .X;R/ is commutative if, and only if, R is commutative. Suppose that R has an identity; show that Fun .X;R/ has an identity and describe the units of Fun .X;R/ . 6.1.13. Let S ± End K .V / , where V is a vector space over a ﬁeld K . Show that S0 D f T 2 End K .V / W TS D ST for all S 2 S g is a subring of End K .V / . 6.1.14. Let V be a vector space over a ﬁeld K . Let G be a subgroup of GL.V / . Show that the subring of End K .V / generated by G is the set of all linear combinations P g n g g of elements of G , with coefﬁcients in Z . 6.1.15.
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.