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1. An ideal / of a ring R is radical if it satisfies the following property: for all r € R and positive integer n, if 7-11 E I then r € 1. Show that I is radical if and only if R/I has no non-zero nilpotent elements. (A ring with no non-zeero nilpotent elements is called reduced.) 2. Show that for K a field, the only idesals in the ring of TI X 72 matrices, Mn(K), are (0) and M, ,KK. 3. Prove is a subring of R and that it is isomorphic to Q|=|//z2 - 2). 4. Show that (2) is a prime but not maximal ideal in Z|r]. 5. Show that in a finite commutative ring every prime ideal is maximal.

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