6.4. INTEGRAL DOMAINS 289 6.3.9. For any ring R , and any natural number n , we can deﬁne the matrix ring Mat n .R/ consisting of n-by-n matrices with entries in R . If J is an ideal of R , show that Mat n .J/ is an ideal in Mat n .R/ and furthermore Mat n .R/= Mat n .J/ Š Mat n .R=J/ . Hint: Find a natural homomorphism from Mat n .R/ onto Mat n .R=J/ with kernel Mat n .J/ . 6.3.10. Let R be a commutative ring. Show that RŒxŁ=xRŒxŁ Š R . 6.3.11. This exercise gives a version of the Chinese remainder theorem . (a) Let R be a ring, P and Q ideals in R , and suppose that P \ Q D f0 g , and P C Q D R . Show that the map x 7! .x C P;x C Q/ is an isomorphism of R onto R=P ˚ R=Q . Hint: Injectivity is clear. For surjectivity, show that for each a;b 2 R , there exist x 2 R , p 2 P , and q 2 Q such that x C p D a , and x C q D b . (b) More generally, if P C Q D R , show that R=.P \ Q/ Š R=P ˚ R=Q . 6.3.12. (a) Show that integers m and n are relatively prime if, and only if, m Z C n Z D Z if, and only if, m Z \ n Z D mn Z . Conclude that if m and n are relatively prime, then Z mn Š Z m ˚ Z n as rings. (b) State and prove a generalization of this result for the ring of poly-
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