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Unformatted text preview: 6.2. HOMOMORPHISMS AND IDEALS 281 6.2.6. Let S be a subset of a set X . Let R be the ring of realvalued functions on X , and let I be the set of realvalued functions on X whose restriction to S is zero. Show that I is an ideal in R . 6.2.7. Let R be the ring of 3-by-3 upper triangular matrices and I be the set of upper triangular matrices that are zero on the diagonal. Show that I is an ideal in R . 6.2.8. Show that if R is a ring with identity element and x 2 R , then Rx D f rx W r 2 R g is the principal left ideal generated by x Similarly, xR D f xr W r 2 R g is the principal right ideal generated by x . 6.2.9. Show that if R is a ring with identity, then the principal ideal gener- ated by x 2 R is .x/ D f a 1 xb 1 C a 2 xb 2 C C a n xb n W n 2 N ;a i ;b i 2 R g : 6.2.10. Show that any field is a simple ring. 6.2.11. Show that the ring M of n-by- n matrices over R has no ideals other than and M . Conclude that any ring homomorphism ' W M ! S is either identically zero or is injective.is either identically zero or is injective....
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- Fall '08